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Los  Angeles 


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This  book  is  DUE  on  the  last  date  stamped  below 


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FOR    COLLEGIATE    STUDENTS    OF 

AGRICULTURE  AND   GENERAL  SCIENCE 
REVISED  EDITION 


A  SERIES   OF   MATHEMATICAL  TEXTS 

EDITED   BY 
EARLE  RAYMOND  HEDRICK 

THE  CALCULUS 

By  ELLERY  WILLIAMS  DAVIS  and  WILLIAM  CHAKLES  BRENKE. 

ANALYTIC  GEOMETRY  AND  ALGEBRA 

By  ALEXANDER  ZIWET  and  Louis  ALLEN  HOPKINS. 

ELEMENTS  OF  ANALYTIC  GEOMETRY 

By  ALEXANDER  ZIWET  and  Louis  ALLEN  HOPKINS. 

PLANE        AND        SPHERICAL        TRIGONOMETRY        WITH 

COMPLETE  TABLES 
By  ALFRED  MONROE  KENYON  and  Louis  INGOLD. 

PLANE        AND        SPHERICAL        TRIGONOMETRY        WITH 

BRIEF  TABLES 
By  ALFRED  MONROE  KENYON  and  Louis  INGOLD. 

ELEMENTARY  MATHEMATICAL  ANALYSIS 

By  JOHN  WESLEY  YOUNG  and  FRANK  MILLETS  MORGAN. 

COLLEGE  ALGEBRA 

By  ERNEST  BROWN  SKINNER. 

MATHEMATICS       FOR        COLLEGIATE        STUDENTS        OF 

AGRICULTURE  AND  GENERAL  SCIENCE 
By  ALFRED  MONROE  KENYON  and  WILLIAM  VERNON  LOVITT. 

PLANE  TRIGONOMETRY 

By  ALFRED  MONROE  KENYON  and  Louis  INGOLD. 

THE  MACMILLAN  TABLES 

Prepared  under  the  direction  of  EARLE  RAYMOND  HEDRICK. 

PLANE  GEOMETRY 

By  WALTER  BURTON  FORD  and  CHARLES  AMMERMAN. 

PLANE  AND  SOLID  GEOMETRY 

By  WALTER  BURTON  FORD  and  CHARLES  AMMERMAN. 

SOLID  GEOMETRY 

By  WALTER  BURTON  FORD  and  CHARLES  AMMERMAN. 

CONSTRUCTIVE  GEOMETRY 

Prepared  under  the  direction  of  EARLE  RAYMOND  HEDRICK. 

JUNIOR  HIGH  SCHOOL  MATHEMATICS 

By  WILLIAM  LEDLEY  VOSBURGH  and  FREDERICK  WILLIAM 
GENTLEMAN. 


MATHEMATICS 


FOB    COLLEGIATE    STUDENTS    OF 


REVISED  EDITION 


BY 

ALFRED   MONROE  KENYON 

PROFESSOR   OF   MATHEMATICS    IN   PURDUE    UNIVERSITY 
AND 

WILLIAM  VERNON  LOVITT 

ASSOCIATE   PROFESSOR    OF    MATHEMATICS   IN    COLORADO    COLLEGE 


Nefa  gorfc 
THE   MACMILLAN   COMPANY 

LONDON:  MACMILLAN  &  CO.,  LTD. 

1918 


COPYBIQHT,    1917, 

BY  THE   MACMILLAN   COMPANY. 


Set  up  and  electrotyped.     Published  December,  1917. 


Nortoooli 
Printed  by  Berwick  &  Smith  Co.,  Norwood,  Mass.,  U.S.A. 


ft  4- 2 


PREFACE 

This  book  is  designed  as  a  text  in  freshman  mathematics  for 
students  specializing  in  agriculture,  biology,  chemistry,  and 
physics,  in  colleges  and  in  technical  schools. 

The  selection  of  topics  has  been  determined  by  the  definite 
needs  of  these  students.  An  attempt  has  been  made  to  treat 
these  topics  and  to  select  material  for  illustration  so  as  to  put 
in  evidence  their  close  and  practical  relations  with  everyday 
life,  both  in  and  out  of  college.  It  is  certain  that  the  interest 
of  the  student  can  be  aroused  and  sustained  in  this  way.  We 
believe  also  that  he  can  be  trained  to  understand  and  to  solve 
those  mathematical  problems  which  will  confront  him  in  the 
subsequent  years  of  his  college  work  and  in  after-life,  without 
losing  anything  in  orderly  arrangement  or  in  clear  and  accurate 
logical  thinking. 

Reference  to  the  table  of  contents  will  indicate  the  scope  and 
proportions  of  the  material  presented  and  something  of  the 
means  employed  in  relating  the  material  to  the  vital  interests 
of  the  student  and  of  correlating  it  to  his  experience  and  his 
intellectual  attainments.  Many  of  the  chapter  subjects  and 
paragraph  headings  are  traditional.  Nothing  has  been  intro- 
duced merely  for  novelty.  Since  this  course  is  to  constitute  the 
entire  mathematical  equipment  of  some  students,  some  chapters 
have  been  inserted  which  have  seldom  been  available  to  fresh- 
men; for  example,  the  chapters  on  annuities,  averages,  and 
correlation,  and  the  exposition  of  Mendel's  law  in  the  chapter 
on  the  binomial  expansion. 

Particular  attention  has  been  given  to  the  illustrative  examples 
and  figures,  and  to  the  grading  of  the  problems  in  the  lists. 
The  exercises  constitute  about  one  fifth  of  the  text  and  contain 


yi  PREFACE 

a  wealth  of  material.  They  include  much  data  taken  from 
agricultural  and  other  experiments,  carefully  selected  to  stimu- 
late thinking  and  to  show  the  application  of  general  principles 
to  problems  which  actually  arise  in  real  life,  and  in  the  solution 
of  which  ordinary  men  and  women  are  vitally  interested. 

The  book  is  intended  for  a  course  of  three  hours  a  week  for 
one  year,  but  it  can  be  shortened  to  a  half-year  course.  The 
chapters  on  statics,  small  errors,  land  surveying,  annuities, 
compound  interest  law,  and  as  many  as  is  desired  at  the  end, 
can  be  omitted  without  breaking  the  continuity  of  the  course. 

The  first  two  chapters  are  more  than  a  mere  review.  This 
matter  is  so  presented  as  to  give  the  student  a  new  point  of 
view.  The  treatment  will  show  the  significance  and  importance 
of  certain  fundamental  relations  among  the  concepts  and 
processes  of  arithmetic  and  algebra  which  the  student  may 
have  used  somewhat  mechanically  .in  secondary  school  work. 
Well  prepared  students  can  read  these  chapters  rather  rapidly, 
however. 

The  four  place  mathematical  tables  printed  at  the  end  of  the 
text  have  been  selected  and  arranged  for  practical  use  as  the 
result  of  long  experience  and  actual  use  in  computing,  and  are 
adapted  to  the  requirements  of  the  examples  and  exercises  in 
the  book. 

The  first  edition  of  this  book  contained  problems,  formulas 
and  other  matter  taken  from  a  large  number  of  sources.  Those 
passages  that  were  directly  from  other  books  have  now  been 
entirely  rewritten ;  but  the  book  remains  indebted  to  a  num- 
ber of  others,  notably  SKINNER,  Mathematical  Theory  of  Invest- 
ment, and  DAVENPORT,  Principles  of  Breeding.  Other  references 
occur  throughout  the  text. 

A.  M.  KENYON, 
W.  V.  LOVITT. 


CONTENTS 

CHAPTER                                                                                                                                                       PAGES  ARTICLE 

I.  Introduction 1-10  1 

II.  Review  of  Equations 11-35  12 

III.  Graphic  Representation 36-71  34 

IV.  Logarithms 72-90  63 

V.  Trigo'nometry 91-138  74 

VI.  Land  Surveying 139-152  108 

VII.  Statics 153-176  122 

VIII.  Small  Errors 177-189  139 

IX.  Conic  Sections 190-217  149 

X.  Variation .  .> 218-225  167 

XI.  Empirical  Equations 226-242  173 

XII.  The  Progressions 243-253  178 

XIII.  Annuities 254-261  186 

XIV.  Averages 262-268  194 

XV.  Permutations  and  Combinations 269-274  201 

XVI.  The     Binomial     Expansion  —  Laws     of 

Heredity 275-285  207 

XVII.  The  Compound  Interest  Law 286-290  217 

XVIII.  Probability 291-303  219 

XIX.  Correlation 304-313  232 

TABLES 314-333 

INDEX;.  . 335-337 


vu 


MATHEMATICS 


CHAPTER  I 
INTRODUCTION 

1.  Uses  of  Mathematics.     The  applications  of  mathematics 
are  chiefly  to  determine  the  magnitude  of  some  quantity  such 
as  length,  angle,  area,  volume,  mass,  weight,  value,  speed,  etc., 
from  its  relations  to  other  quantities  whose  magnitudes  are 
known,  or  to  determine  what  magnitude  of  some  such  quantity 
will  be  required  in  order  to  have  certain  prescribed  relations 
to  other  known  quantities. 

2.  Measurement.     To  measure  a  quantity  is  to  find  its  ratio 
to  a  conveniently  chosen  unit  of  the  same  kind.     This  number 
is  called  the  numerical  measure  of  the  quantity  measured. 

•  The  expression  of  every  measured  quantity  consists  of  two 
components:  a  number  (the  numerical  measure),  and  a  name 
(that  of  the  unit  employed).  For  example,  we  write:  10  inches, 
27  acres,  231  cubic  inches,  16  ounces,  22  feet  per  second. 

3.  Arithmetic  and  Algebra.     In  arithmetic   we  study  the 
rules  of  reckoning  with  positive  rational  numbers.     In  algebra 
negative,   irrational,   and  imaginary  numbers  are  introduced, 
letters  are  used  to  represent  classes  of  numbers,  and  the  rules 
of  reckoning  are  extended  and  generalized.     Algebra  differs  from 
arithmetic  also  in  making  use  of  equations  for  the  solution  of 
problems  requiring  the  discovery  of  numbers  which  shall  satisfy 
certain  prescribed  conditions. 

2  1 


2  MATHEMATICS  [I,  §4 

4.  Positive  Numbers.  The  natural  numbers  1,  2,  3,  4,  etc., 
are  the  foundation  on  which  the  whole  structure  of  mathe- 
matics is  built.  They  are  also  called  whole  numbers,  or 
positive  integers.  Together  with  the  fractions,  of  which  1/2, 
5/3,  .9,  2.31,  are  examples,  they  form  the  class  of  positive 
rational  numbers. 

Every  positive  rational  number  can  be  expressed  as  a  fraction 
whose  numerator  and  denominator  are  whole  numbers. 

Two  quantities  of  the  same  kind  are  said  to  be  commensur- 
able when  there  is  a  unit  in  terms  of  which  each  has  for  numer- 
ical measure  a  whole  number.  Consequently,  their  ratio  is 
a  rational  number.  If  two  quantities  are  not  commensurable, 
they  are  said  to  be  incommensurable. 

The  ratio  of  two  quantities  which  are  incommensurable, 
such  as  the  side  and  the  diagonal  of  a  square,  or  the  diameter 
and  the  circumference  of  a  circle,  is  an  irrational  number. 

No  irrational  number  can  be  expressed  as  a  fraction  whose 
numerator  and  denominator  are  whole  numbers.  However,  it 
is  always  possible  to  find  two  rational  numbers,  one  less  and  the 
other  greater  than  a  given  irrational  number,  whose  difference 
is  as  small  as  we  please.  For  example, 

3.162277  <  VlO  <  3.162278 

and  the  difference  between  the  first  and  the  last  of  these  num- 
bers is  only  .000001.  Two  such  rational  numbers  whose  dif- 
ference is  still  less  can  easily  be  found.  In  all  practical  appli- 
cations, one  of  these  rational  numbers  is  used  as  an  approximation 
for  the  irrational  number.  Thus,  we  may  find  the  length  of  the 
circumference  of  a  circle  approximately  by  multiplying  its  di- 
ameter by  3f.  If  a  closer  approximation  is  needed,  the  value 
3.1416  is  often  used. 

The  (positive)  rational  and  the  (positive)  irrational  numbers 
make  up  the  class  of  (positive)  real  numbers. 


I,  §7]  INTRODUCTION  3 

5.  Negative  Numbers.     Zero.     To    every    positive    real 
number  r,  there  corresponds  a  negative  real  number  —  r,  called 
negative  r.     The  negatives  of  the  natural  numbers  are  called 
negative  integers.     The  real  number  zero  separates  the  negative 
numbers  from  the  positive  numbers.     It  is  neither  positive  nor 
negative  and  corresponds  to  itself. 

The  negatives  of  negative  numbers  are  the  corresponding 
positive  numbers;  thus,  —  (—  2)  =2. 

6.  The  Four  Fundamental  Operations.     The  direct  opera- 
tions of  addition  and   multiplication  of  real  numbers   are  so 
defined  that  they  are  always  possible,  and  so  that  the  result 
in  each  case  is  a  unique  real  number.     These  operations  are 
subject  to  the  rules  of  signs  and  to  the  following  fundamental 
laws  of  algebra. 

I.  The  commutative  law: 

a  +  6  =  6  +  a,         ab  =  ba. 

II.  The  associative  law: 

(a  +  &)  +  c  =  a  +  (b  +  c),         (a6)c  =  a(6c). 

III.  The  distributive  law: 

a(b  +  c)  =  ab  +  ac- 

The  indirect  operations  of  subtraction  and  division  of  real 
numbers  are  always  possible,  division  by  zero  excepted,*  and 
the  result  is  a  unique  real  number. 

7.  Involution  and  Evolution.     Involution,  or  raising  to  pow- 
ers, is  always  possible,  and  the  result  is  unique  when  the  base 
is  any  real  number  provided  the  exponent  is  a  positive  integer. 

*  Division  by  zero  is  excluded  because,  in  general,  it  is  impossible,  and  when  possible 
it  is  trivial.  Thus  there  is  no  real  number  which  will  satisfy  the  equation  0  -x  =  o  4= 
0,  and  every  real  number  satisfies  the  equation  0  -x  =  0. 


4  MATHEMATICS  [I,  §7 

Evolution,  or  extraction  of  roots,*  is  not  always  possible. 
Even  when  possible,  it  is  not  always  unique.  In  particular, 
the  square  of  every  real  number  is  a  positive  real  number. 
Hence  no  negative  number  can  have  a  real  square  root.  On 
the  other  hand,  every  positive  real  number,  a,  has  two  real 
square  roots:  a  positive  one,  which  is  denoted  by  the  symbol 
•^a;  and  a  negative  one,  which  is  denoted  by  —  Va.  In  fact, 
every  positive  real  number  has  exactly  two  real  nth  roots  of 
every  even  index  n,  denoted  by  Va  and  —  "N/o,  respectively. 
Every  real  number,  r,  has  a  unique  real  nth  root  of  every  odd 
index  n,  denoted  by  Vr;  it  is  positive  when  r  is  positive,  and 
negative  when  r  is  negative. 

8.  Rational  Exponents.  Involution  is  extended  to  frac- 
tional exponents  as  follows.  If  a  is  a  positive  real  number, 
and  if  m  and  n  are  natural  numbers,  we  define  amln  by  the 
equation 

.         «i — 
amln  —  -\am. 

For  example, 

82/3    =     ^    =    4> 
ni— 

In  particular,  a1'"  =  Va  denotes  the  unique  real  positive  nth 
root  of  a. 

If  r  is  a  positive  rational  number,  (—  a)r  is  defined  only  when 
r,  expressed  as  a  fraction  m/n,  in  its  lowest  terms,  has  an  odd 
denominator  and  in  this  case, 

(-  a)r  =  (-  l)mar. 

For  example,  (-  32)  •«  =  (-  32)3/5  =  (-  1)3(32)3/5  =  -  8,  and 
(-  32)  •«  =  (-  32)4/5  =  (-  1)«(32)4/5  =  16. 

In  particular,  (—  a)1/n  =  —  a1/n  =  —  Va,  if  n  is  odd. 

*  The  index  of  a  root  is  always  a  positive  integer. 


I,  §9]  INTRODUCTION  5 

9.  Negative  and  Zero  Exponents.     By  definition,  we  write 
a~b  =  —b         and         a°  =  1, 

Q 

provided  a  =|=  0.     Thus  a~b  is  defined  for  the  same  real  values 
of  a  and  6  as  is  a6  and  the  two  are  reciprocals.*     For  example, 

1         1  /I  \~5/2  1 

g-2/3  _  _!_  =  L  * 

82/s       4> 


A  consequence  of  this  definition  is  the  rule:  A  factor  may  be 
moved  from  the  numerator  to  the  denominator  of  a  fraction,  or  vice 
versa,  on  changing  the  sign  of  its  exponent.  For  example, 


a26c~3  _  2-1d~1e  _ 
2de~l   "  a-^-'c3  ~  2c3d  ' 

=  (a?  -  z2)-i/2, 


Va2  —  x 

or2  +  2x-*y~l  +  7/-2  =  -  +  —  +  -  . 
xz      xy      y2 

EXERCISES 

1.  Verify  the  fundamental  laws  of  algebra  by  making  use  of  the 
three  numbers  f,  —  5-J-,  |. 

2.  How  many  real  square  roots  has  24.5?     —  4.5? 

3.  How  many  real  cube  roots  has  6f?     —  12£? 

4.  Find  the  numerical  value  of  each  of  the  following  expressions, 
exactly  when  rational,  correct  to  three  decimal  places  when  irrational. 

(a)  9s/2.  (e)  (32)s/5.  (i)  (-0.027)  »/8. 

(*>)  Gfr)2/s.  (/)(-32)«».  (j)  (H)"4- 

(c)  v<3.  (g)  (-2)"'.  (*)  (t)1". 

(d)  (-  2)«».  (A)  (-  0.375)^.  (1)  (-  «)w. 

*  Two  numbers  are  reciprocals  when  their  product  is  +  1.     Every  real  number  has  a 
reciprocal  except  0,  which  has  none. 


6  MATHEMATICS  [I,  §  9 

5.    Write  each  of  the  following  expressions  without  radical  signs. 
(a)   -^32.  (b)   Vl28.  _c) 


6)2.  (i)    ^4?-84.  0')    -v-^ 

6.   Write  each  of  the  following  expressions  without  negative  expo- 
nents and  simplify  when  possible. 


(27  x-V2"12)~1/3-       (c)    (a^  +  fc-2)-1.  (/)    (a 


ar1  +  y-1  a"1  -  fe-1  a;-1?/0  +  x°y 

,..    30a-W  -S-^tfb0  ffcx    x-ly-*z~3  +  x*y*z 

32a°6-1+3a6    '  ar3?/-^-1  +  XT/V  ' 

10.  Laws  of  Exponents.    The  following  five  laws  are  useful 
for  the  reduction  of  exponential  and  radical  expressions  to  sim- 
pler forms.     They  are  valid,  (1)  when  the  bases  are  any  real 
numbers  whatever,  provided  the  exponents  are  integers  or  zero, 
and  (2)  when  the  exponents  are  any  real  numbers  whatever, 
provided  the  bases  are  positive. 

I.  ab  -  ac  =  ab+c. 

EXAMPLES.    32  •  tr*  =  3~2.  (  -  |)B(  -  f  )-3  =  (  -  f)2. 

(2)-l/3(|)l/2    =    (!)l/6-  g-l/3  .  g2    =  85/3. 

11.  acbc  =  (ab)c. 

EXAMPLES.     2353  =  103.  (-  3)-2(-  5)~2  =  (15)~2. 

(17)1/3(^)1/3    =   51/8, 

III.  (ab)c  =  abc. 

EXAMPLES.     (28)2  =  26.  [(-  |)-2]-3  =  (-  f)6. 

[(f)1/3l6  =  (I)2- 


I,  §11]  INTRODUCTION 

ab 

IV.  r  =  ab~c. 
ac 

Q2  (_    5N1/2 

EXAMPLES.     —  =  3'.         (_  |j-.,.  =  (-  f>"°- 

V.  £. 

4-2/5  (2\-2 

T?YAU*T>ri?a  — —    f4\-2/5  v"x         _    f2\-2 

EXAMPLES.          3_2/5   -    ($)         .  3^_2  -    UJ      . 

These  laws  are  readily  proved  when  the  exponents  are  positive 
integers.  Thus,  to  prove  law  II,  when  the  exponent  is  a  positive 
integer  n,  we  write 

(1)(2)(3)  (n)(n  (2)  (3)  (n) 

an-bn  =  a- a- a  •  •  •  a-b-b-b  •  •  •  b 

(1)     (2)     (3)  (n) 

=  ab-ab-ab  •  •  •  ab  =  (ab)n. 

Similarly,  each  of  the  other  laws  can  be  proved  when  the 
exponents  are  positive  integers.  When  the  exponents  are 
negative,  we  make  use  of  the  definition  of  §  9.  If  they  are 
positive  fractions  we  make  use  of  the  following  lemma:  //  a 
and  b  are  real  numbers  of  like  sign,  and  if  an  =  bn,  where  n  is  a 
positive  integer,  then  a  =  b. 

11.  Binomial  Theorem.  By  multiplying  out,  we  find  the 
following  equalities: 

(x  +  7/)2  =  x1  +  2#y  +  ?/2, 

(x  +  y)3  =  x3  +  3z2?/  +  3z?/2  +  y3, 

(x  +  2/)4  =  a;4  +  4:X3y  +  6.r27/2  +  4xy3  +  y4, 

(x  +  yY  =  x5  +  5x4y  +  lOo;3?/2  +  Wx2y3  +  5xy4  +  y5. 

By  observing  the  coefficients  and  the  exponents  of  x  and  of  y 
in  the  various  terms,  we  observe  the  law  by  which  these  results 
can  be  written  down  without  the  work  of  multiplying  them  out. 

In  the  expansion  of  (x  +  y)n  for  n  —  2,  3,  4,  5,  we  note  the 
following  facts: 

(1)  The  number  of  terms  is  n  +  1- 


8  MATHEMATICS  [I,   §11 

(2)  The  exponent  of  x  in  the  first  term  is  n  and  it  decreases 
by  1  in  each  succeeding  term;  the  exponent  of  y  in  the  second 
term  is  1  and  it  increases  by  1  in  each  succeeding  term. 

(3)  The  first  coefficient  is  1;  the  second  is  n;  the  coefficient 
of  any  term  after  the  second  may  be  found  from  the  preceding 
term  by  multiplying  the  coefficient  by  the  exponent  of  x  and  dividing 
by  a  number  1  greater  than  the  exponent  of  y. 

These  three  statements  constitute  the  binomial  theorem, 
which  will  be  proved  in  §  208,  Chapter  XVI,  for  all  values  of 
x  and  y  no  matter  how  large  the  positive  integer  n  may  be. 
The  coefficients  which  appear  in  these  expansions  are  called 
binomial  coefficients.  For  example,  the  numbers 

1,     5,     10,     10,     5,     1 

are  the  binomial  coefficients  for  the  fifth  power.  The  binomial 
coefficients  for  the  second,  third,  fourth,  and  fifth  powers  should 
be  memorized. 

EXERCISES 

Use  the  laws  of  exponents  to  combine  and  simplify  the  following 
expressions. 

1.  g-^-S^-S-^-S2  -j-  83/4-81/12.       2.  32/B-42/B-52/B  4-  152/B-82/5. 
3.  (32-31/2-56/2)2  -i-  (73  -  102).          4.  (11  -32  +  74)1/2. 

C3/4  1  03/2 

K      (K4   _   92.96^1/2  A      °  _  7      •*"* 

D<  88'12'  33/2  ' 

402"  V48  V54  Vl2 

O.      iro/o    •  a.         ;r~  .  1U.    A   -  .  11.    —  —  . 

52/3  V3  M36  >/6 


12. 


I,  §  11]  INTRODUCTION  9 

Perform  the  indicated  operations  and  simplify  each  of  the  following 
expressions  when  possible. 


\ 


16.  .  7       •ax-  18 

\Wy)  UW  ' 

19.    f"-l™-*Yl.  20  f^blY3'  21 

\    *V2    /  '  \8aV/ 

22     /     x  +  2     \-i.  /r«±&V 

\x*+x-2j  •    \a-b)    ' 

24.   63a4x5  -^  9a3x2  H-  3a2x.  25.    (o°  +  6)  (a  +  6°). 

' 


3a262 


28 


30     2a2  +  7ax  +  3x2  ^  3a2  +  7ax  +  2x2 


2a  +  x  a  +  3x 


31 


a2  -  a  -  20     a2  -  2a  -  15         a2  -  a  a2 

33. 


x2  -  5x  +  4     x2  -  lOx  +21      x2  -  9x  +  20 

4.         a  +  3        va2  +a-2  .  a2  +3a+2 


6 
Multiply : 

34.  a5/6  _ 

35.  a1/2  +  2&1/2  -  3c1/2  by  a1/2  -  261/2  +  3c1/2. 

36.  x4/3  +  2x  +  3x2/3  +  2x1/3  +  x°  by  x2/3  -  2X1/3  +  x°. 

37.  Vx3  -  x2y  -  XT/2  +  y3  by  Vx3  +  3x2y  +  3xy2  +  y3. 

38.  o1/4  -  61/4  by  a3/4  +  &3/4.  39.   x3/5  -  ?/2/8  by  x2/5  +  y3'*. 

40.  (a1/2^/2  +  c1/2)2  by  (a1/2*)1/2  -  c1/2)2. 

41.  (a-1/2  -  3)2  by  (3O1/2  +  I)2. 
Divide : 

42.  x8/2  +  x2  -  2X1/2  +  1  by  x  +  x1/2  -  1. 

43.  x3  +  27x  -  9x1/2  -  10  by  x  -  3x1/2  +  5. 

44.  x-y-  6x2/3  +  12X1/3  -  8  by  x1/3  -  y1'3  -  2. 

45.  a6/2  -  a26  +  a3  2c  -  oc  +  a1/2&  -  1  by  o1/2  -  1. 

46.  a2  +  Sa1/4  +  7  by  a1/2  +  2o1/4  +  1. 


10  MATHEMATICS  [I,  §  11 

Reduce  each  of  the  following  to  its  simplest  form  : 
47.    Vl2.25zV.  48.    v/15.625a669. 


49.  V343a10625.  50.    V3a26  -  2a2c. 

51.  V(a3  +  53)  (02  _|_  52  _  atyf  52.    Vx4?/2  -  2xy  +  x2?/4. 

53.  VVl024.  54. 

55.  V27av/27a64. 

57.  •\/1.35a2V6.25a;!. 

59.  V|05 -2V605+V845.  60.    *V  192  -  2v/375  +  #648 

61.  V72-V8-V50.  62.    ^81  -  2\yl92  +  ^375. 

63.  (VI53  -  VTI7  + A/52  -  V68)(V/5T  + V39). 

64.  (Vl2  +  V3  +  Vs})(vls 

65.  (2  +  V3  +  v/4)(2  -  V3 

66.  (3V20 -4V5  +  5V2 -3V8)(V5  + VCL5). 


67.    V19  +  3V2-  V19-3V2.          68.    Vl6  + Vl3- 
Rationalize  the  denominator  of  each  of  the  following  fractions : 

69.    -Iz^L  70.    1^.  71.    - 

3-2V2  2+V5  2V5+3V2 


72     x/3+v/2  +  \/2  -V3,  73     2+V5x5  -V2. 

\/3-V2      V2+V3  2-Vg      5+V2 

74     V2  -  \73  75     Vl89+3V20_ 


V3^_|_V48-V'50-v/75  V84-V80 

Expand  each  of  the  following  expressions : 
76.    (2p 
79.    (1  - 

82. 

85.  (1  +  x2)2. 

88.  (fc2+3)2. 

91.  (Va+Va; 

94.  (2x  -  37/)3 

97.  (1  +  x)3. 
100. 


77. 

(5c  -  9d)2. 

78. 

(4m  -  3n)2. 

80. 

(1  -  *)2. 

81. 

(fa  +  f  6)2. 

83. 

(l-i)2- 

84. 

(^  -  tl/)2. 

86. 

(1  -  X2)2. 

87. 

(1  +Vx)2. 

89. 

(2t2  +  5)2. 

90. 

(a2  +  a6)2. 

92. 

(a-l/2  +  xl/t)». 

93. 

(ftl/3  _  2/l/2)2_ 

95. 

(a  +  ^b)3. 

96. 

(v^+Vm) 

98. 

(1  -  x)3. 

99. 

(1  +  x2)3. 

101. 

(x+y-  a)2. 

102. 

(a2  +  ab  +  W 

CHAPTER  II 
REVIEW   OF  EQUATIONS* 

12.  Use  of  Equations.     As  indicated  before,  the  chief  ad- 
vantage of  algebra  over  arithmetic  in  solving  problems  lies  in 
the  method  of  attack.     The  algebraic  method  is  to  translate 
the  problem  into  an  equation  and  then  to  solve  the  equation  by 
general  methods. 

13.  Definition  of  an  Equation.     An  equation  is  a  statement 
of  the  equality  of  two  expressions.     Each  of  the  expressions 
may   contain  letters   and   figures   called   knowns,   representing 
numbers  supposed  to  be  given  or  known;  letters  called  unknowns, 
representing  numbers  to  be  found;  and  symbols  of  operation 
and  combination,  such  as  +,  — ,  etc. 

As  examples  of  equations  in  one  unknown,  we  may  write 

(1)  x  +  13  =  2x  -  7, 

(2)  x(S  -  x)  =  2(x  +  l)(z2  -  x  +  1), 

(3)  x(x  +  2)  =  (x  -  l)(x  -  2)  +  5x  -  2, 

2x  +  l      2x  -  1  _ 
x-  1        x  +  l 

(5)  7  Vx  —  6  +  6  >/3x  +  4  =  4x  +  3. 

As  examples  of  equations  in  two  unknowns,  we  may  write 

(6)  x2  -  if  +  2y  =  1, 

(7)  (2x  —  ?/)2  —  5x2  =  5?/2  —  (x  +  2y)z. 
Similarly,  we  may  have  equations  in  more  than  two  unknowns. 

*This  chapter  is  intended  for  review  work.  Parts  of  it  may  bo  omitted  at  the  dis- 
cretion of  the  instructor,  if  it  appears  that  the  students  do  not  need  to  review  some  of 
the  topics. 

11 


12  MATHEMATICS  [II,  §13 

The  expression  on  the  left  of  the  equality  sign  is  called  the 
left  member,  or  the  left  side,  of  the  equation.  The  other  is  called 
the  right  member,  or  right  side. 

14.  Substitution.     It  is  often  necessary  to  substitute  for  the 
unknowns  in  an  expression  such  as  one  of  the  members  of  the 
above  equations,  certain  definite  numbers,  called  values  of  the 
unknowns.     The  result  of  such  substitution  is,  in  general,  to 
reduce  the  expression  to  a  single  number. 

Thus,  if  we  put  10  for  x  in  equation  (1),  the  left  side  reduces  to  23 
and  the  right  side  to  13.  If  we  put  20  for  x,  each  member  reduces  to 
the  same  number,  33. 

Again,  if  we  put  1  for  x  and  1  for  y  in  equation  (6),  the  left  side 
reduces  to  2  and  the  right  side  to  1;  but  if  we  put  2  for  x  and  —  1  for 
y,  each  member  reduces  to  1. 

15.  Solution  of  an  Equation.     Any  set  of  values  of  the  un- 
knowns which  reduces  each  of  the  two  members  of  an  equation 
to  the  same  number  is  said  to  satisfy  the  equation,  and  to  be  a 
solution  of  the  equation.     A  solution  of  an  equation  in  one 
unknown  is  also  called  a  root  of  the  equation. 

The  final  test  to  determine  whether  a  set  of  values  of  the 
unknowns  in  an  equation  is  a  solution  or  not,  is  to  substitute 
these  values  for  the  unknowns  and  see  whether  the  equation 
is  satisfied  or  not. 

For  example,  x  =  20  is  a  solution  of  equation  (1),  §  13.  The  value 
x  =  10  does  not  satisfy  it.  Again,  z  =  |,  x  =  1,  x  =  —  2,  are  three 
solutions  of  (2).  Every  real  number  is  a  solution  of  (3).  The  value 
x  =  2  is  a  solution  of  (4).  The  value  x  =  15  is  a  solution  of  (5).  The 
values  x  =  2,  y  =  —  1  constitute  a  solution  of  (6).  Every  pair  of  real 
numbers  constitutes  a  solution  of  (7). 

16.  Identities.     An  equation  which  is  satisfied  by  all  values 
of  the  unknowns  (excepting  those  values  if  there  are  any  for 
which  either  member  is  not  defined)  is  called  an  identity.     An 


II,  §  16]  REVIEW  OF   EQUATIONS  13 

equation  which  is  not  an  identity  is  called  a  conditional  equation, 
or  when  no  ambiguity  is  likely  to  arise,  simply  an  equation. 

EXAMPLES.     Of  the  equations  in  §  13,  (3)  and  (7)  are  identities,  the 
others  are  conditional  equations.     Also, 


is  an  identity;  it  is  satisfied  by  all  values  of  x,  except  x  =  1  for  which 
neither  side  is  defined. 

The  distinction  in  point  of  view  between  identities  and  con- 
ditional equations  is  fundamental.  To  show  that  an  equation 
is  not  an  identity,  we  need  only  find  a  single  set  of  values  of 
the  unknown  quantities  for  which  both  sides  are  defined,  and 
for  which  the  equation  is  not  true. 

EXERCISES 

1.  Which  of  the  numbers    —  3.5,    —  2,    —  1,  0,  \,  2,  satisfy  the 
equation 

ix+«±2_      10      ? 
3      ~2x~+~l{ 

2.  Which  of  the  numbers  TV  V7,  2  +    V3,  Vl4,  2  —  V3,  are  solutions 
of  the  equation  x2  +  1  =  4x? 

3.  Which  of  the  following  pairs  of  numbers  (0,  0),   (1,  3),   (4,  2), 
(0,  2),  (1,  -  1),  (3,  -  1),  (4,  0),  (3,  3),  satisfy  the  equation 


Is  this  equation  an  identity? 

4.  Which  of  the  following  pairs  of  numbers  (0,  1),  (1,  1),  (—  1,  0), 
(2,  3),  (—  2,  1),  (1,  —  1),  (3,  —  2),  are  solutions  of  the  equation 

*  +  2y  =  1+V(1  __  V_  \  ? 

x  +  y  x\         x  +  y)  ' 

Is  this  equation  an  identity? 


14  MATHEMATICS  [II,  §16 

5.  Which  of  the  following  equations  are  identities? 

(a)  x(x2  —  y2)  =  (x  +  y)(x2  —  xy). 
(6)  x(x2  +  y2)  =  (x  -  y)(x2  +  xy). 

(c)  x(x  +  7)  -  (x  +  3)(x  +  4)  +  12  =  0. 

(d)  x(7  -  x)  +  (3  -  x)(4  -  x)  =  12. 

(c)   4x2  +  7x  +  2y  =  0.  (/)  4x2  +  7x  -  2y  =  0. 

(g)  x*  =  (x2  +  l)(x  +  l)(x  -  1)  +  1. 

(h)  tf  =  (1  +  x2)(l  +  x)(l  -x)  +  1. 

(i)    (ax  -  b)2  +  (6x  +  a)2  =  (a2  +  62)(1  +  x2). 

0")    (ax  -  6)2  +  (ax  +  &)2  =  (a2  +  62)(1  +  x2). 

(fc)   (x  -  t/)3  +  (y  -  zY  +  (z-  X?  =  3(x  -  y)(y  -  2) (a  -  x). 

(0    (x  +  2/  +  z)3  -  (x3  +  y3  +  0s)  =  3(x  +  y)(y  +  2) (2  +  x). 

fm)  V* + *^      .  + W  =  1 

^    '  (x  -  y)(x  -  z)  ^  (y  -  z)(y  -  x)  ^  (z  -  x)(z  -  y) 

17.  Equivalent  Equations.     Two  equations  are  said  to  be 
equivalent  when  every  solution  of  the  first  is  a  solution  of  the 
second  and  conversely,  every  solution  of  the  second  is  a  solu- 
tion of  the  first. 

For  example,  the  equations 

5-5  =  o 
3      7 
and 

7x  =  15 

are  equivalent;  each  has  the  unique  solution  x  =  2f. 
On  the  other  hand 

2x  -  3  =  x  -  1 
and 

(2x  -  3)2  =  (x  -  l)z 

are  not  equivalent;  the  latter  has  the  solution  1^,  which  does  not  satisfy 
the  first. 

18.  Transformations  of  Equations.     The  following  changes 
in  an  equation  lead  always  to  an  equivalent  equation: 

1.  Transposition  of  terms  with  change  of  sign. 

2.  Multiplication,  or  division,  of  all  the  terms  by  the  same 
constant  (not  zero). 


II,  §18]  REVIEW   OF   EQUATIONS  15 

If  all  the  terms  of  an  equation  be  transposed  to  the  left  side 
(so  that  the  right  member  is  zero),  if  the  left  member  be  factored, 
and  if  each  of  the  factors  be  equated  to  zero,  then  the  solutions 
of  the  separate  equations  so  formed  are  all  solutions  of  the 
original  equation,  and  it  has  no  others. 

EXAMPLE.     The  equations 

r3  -4-  "vr 
'-~--=x*-x  +  l     and     (x  -  l)(x  -  2)(x  -  3)  =  0 

are  equivalent,  and  the  solutions  of  the  latter  are  seen  by  inspection  to 
be  z  =  1,  x  =  2,  x  =  3. 

The  following  changes  in  an  equation  lead  to  a  new  equation 
which  is  satisfied  by  every  solution  of  the  given  equation,  but 
which  generally  has  other  solutions  also. 

3.  Multiplying    through    by    an    expression    containing    un- 
knowns (defined  for  all  values  of  these  unknowns). 

4.  Squaring  both  members,  or  raising  both  members  to  the 
same  positive  integral  power. 

Since  the  new  equation  is  not,  in  general,  equivalent  to  the 
given  equation,  it  is  necessary  to  test  all  results  by  substituting 
them  in  the  given  equation  in  its  original  form. 

EXAMPLES.     Every  solution  of  the  equation 

x2        1  _  5x 
6~H         ~  6" 

is  a  solution  of  the  equation 

x3  +  6z  =  5x*, 

which  is  formed  by  multiplying  the  first  through  by  6x;  but  they  are 
not  equivalent,  since  x  =  0  satisfies  the  second  but  does  not  satisi'y 
the  first. 

Every  solution  of  the  equation 

3x  -  1  =x  -  1 
x  +  1    ~  x  -  2 


16  MATHEMATICS  [II,  §18 

is  a  solution  of  the  equation 

3x2  -  7x  +  2  =  x2  -  1, 

which  results  from  clearing  the  former  of  fractions.  These  two  equa- 
tions are  in  fact  equivalent.  Each  is  satisfied  by  x  —  |,  and  by  x  —  3, 
and  by  these  only. 

Every  solution  of  the  equation 

x  -  4  =  Vz  +  2 
is  a  solution  of  the  equation 

x2  -  9z  +  14  =  0, 

which  results  from  squaring  and  transposition  in  the  former;  but  they 
are  not  equivalent;  the  latter  equation  has  the  two  solutions  x  =  2, 
x  =  7,  while  the  former  has  only  one,  x  =  7. 

19.  Simultaneous  Equations.  When  a  common  solution  of 
two  or  more  equations  is  sought,  the  equations  are  said  to  be 
simultaneous.  For  example,  each  of  the  equations 

(8)  3z  -  2y  =  4 
and 

(9)  2x  -  y  =  3 

has  an  infinite  number  of  solutions:  (0,  —  2),  (2,  1),  (4,  4), 
(G,  7),  etc.,  satisfy  (8),  and  (1,  -  1),  (2,  1),  (3,  3),  (4,  5),  etc., 
satisfy  (9).  But  (2,  1)  is  the  only  common  solution. 

By  a  solution  of  a  set  of  equations  is  meant  a  common  solu- 
tion of  all  the  equations  of  the  set,  regarded  as  simultaneous 
equations.  Thus,  the  set  of  equations  (8)  and  (9)  has  a  unique 
solution,  namely,  x  =  2,  y  =  1. 

Two  sets  of  simultaneous  equations  are  equivalent  when 
each  set  is  satisfied  by  all  of  the  solutions  of  the  other  set. 

If  each  of  two  or  more  equations  from  a  set  of  simultaneous 
equations  be  multiplied  through  by  any  constant,  or  by  any 
expression  containing  unknowns,*  and  if  the  resulting  equations 

*  Defined  for  all  values  of  the  unknowns. 


II,  §20]  REVIEW   OF  EQUATIONS  17 

be   added   or  multiplied  together,   the  new  equation   will   be 
satisfied  by  all  the  (common)  solutions  of  the  given  set. 
EXAMPLE.     If  in  the  set  of  simultaneous  equations, 

2x2  +  2y2  -  3x  +  y  =  9, 
3x2  +  3yi  +  x  _  y  =  14> 

we  multiply  the  first  by  —  3,  the  second  by  2,  and  add,  the  resulting 
equation 

llx  -  5y  =  1 

is  satisfied  by  every  solution  of  the  given  set.     One  such  solution  is 

x  =  1,  y  =  2. 

20.  Elimination.  By  a  proper  choice  of  multipliers  we  can 
use  the  above  principle  to  secure  a  new  equation  lacking  a 
certain  term,  or  certain  terms,  which  occur  in  the  given  set  of 
equations.  The  missing  terms  are  said  to  have  been  eliminated 
and  this  process  is  called  elimination  by  addition. 

EXAMPLES.     We  can  eliminate  the  term  in  x2  from  the  equations, 


-  2 
5 


5x2  -  9x  =  2, 


2x2  -    x  =  6, 

by  multiplying  by  —  2  and  +  5,  respectively,  and  adding.     The  result 
is  13x  =  26.     We  conclude  that  if  the  given  equations  have  a  common 
solution,  it  is  x  =  2,  and  we  verify  that  this  is  a  solution  of  each. 
If  we  eliminate  x2  from  the  equations, 

-  2j|5x2  +  9x  =  2, 
5||2z2  +  5x  =  5, 
we  obtain 

7x  =  21. 

Since  x  =  3  is  not  a  solution  of  the  given  equations,  they  have  none. 
When  y  is  eliminated  from  the  equations, 

2||3x2  -  4x  -  15y  +  1  =0, 
-  3  ||  2x2  -  3x  -  10y  +  1  =0, 
the  result  is 

x  -  1  =  0 

and  on  substituting  x  =  1  in  either  of  the  given  equations,  we  find 
y  =  0.     Therefore  (1,  0)  is  the  unique  common  solution. 
3 


18  MATHEMATICS  [II,   §20 

When  it  is  possible  to  solve  one  of  a  set  of  simultaneous  equa- 
tions for  one  of  the  unknowns,  we  can  eliminate  this  unknown 
by  substituting  the  value  thus  found  in  the  other  equations  of 
the  set.  This  is  called  elimination  by  substitution. 

For  example,  to  eliminate  t  from  the  set  of  equations, 
x  =  a(l+  <2), 
y  =  o(l  +  0, 
solve  the  second  for  t  and  substitute  this  value  in  the  first.     The  result  is 

x  =  ^  -  2y  +  2a, 

which  is  equivalent  to  the  equation 

y2  —  ax  —  lay  +  2a2  =  0. 

If  we  can  solve  each  of  two  simultaneous  equations  for  the 
same  unknown,  this  unknown  will  be  eliminated  by  equating 
these  two  values  to  each  other.  This  is  called  elimination  by 
comparison. 

Thus,  if  we  solve  each  of  the  equations 

z2  -  xy  -  4x  +  2y  +  1  =0, 
2x*  -  2xy  +  3x  -  2y  +  3  =0, 
for  y,  and  equate  these  values,  the  result  is 

x*  -  4x  +  1  =  2x*  +  3x  +  3 
x  -2  2x  +  2 

which  is  equivalent  to  the  equation 

(5z  +  8)(x  -  1)  =  0. 

21.  Linear  Equations.  An  equation  of  the  first  degree  in 
the  unknown  quantities  is  called  a  linear  equation.  A  set  of 
linear  simultaneous  equations  can  be  solved,  if  they  have  a 
solution,  by  successively  eliminating  the  unknowns  until  a  single 
equation  in  one  unknown  is  obtained. 


II,   §21]  REVIEW  OF   EQUATIONS  19 

EXAMPLES. 

3    2x  +  y  =  4, 
1    x  -  3y  =  9. 

Eliminating  y  by  addition,  we  obtain 

7x  +  0-y  =  21. 
Eliminating  x,  we  get 

Q-x  +  7y  =  -  14. 

We  conclude  that  if  the  given  equations  have  a  solution  it  is  x  =  3, 
y  =  —  2,  and  we  verify  that  this  is  a  solution. 

To  eliminate  x  by  substitution  from  the  equations 

7x  -  Qy  =  15, 
5x  -  Sy  =  17, 

solve  the  first  for  x  and  substitute  this  value  in  the  second.    The 
result  is 

/  n,,.    i     1  c  \ 

-  8y  =  17, 


which  is  equivalent  to  y  =  —  4.  Substituting  —  4  f or  y  in  either  of  the 
given  equations,  we  fird  x  =  —  3.  Finally,  we  verify  that  x  =  —  3, 
y  =  —  4,  is  a  solution  of  the  given  set. 

To  eliminate  x  by  comparison  from  the  equations 

3x  -  7y  =  19, 
2x  -  5y  =  13, 
solve  each  equation  for  x,  and  equate  the  results.    This  gives 

7y  +  19      5y  +  13 
3  2 

which  is  equivalent  to  y  =  —I.  Substituting  this  value  for  y  in  either 
of  the  given  equations  leads  to  x  =  4. 


20  MATHEMATICS  [II,  §21 

EXERCISES 

1.  Solve  the  following  equations  and  determine  whether  or  not  the 
two  equations  in  each  pair  are  equivalent. 

x  +  5      x  +  1  _  x  -  3  _  1      3x  -  7 

(a)  ~2~      T~  ~2x~~  ~  3  ~  ~^x~~ 

n)  y~7 + 2 - y  +  8        2(y ~ 7)    4. ^^ - y  +  3 

()       5  10    '         jf  +  3i,_28  +  y-4~y  +  7' 

3£  -  5  _  9<  -  7  =  2_  5t  +  4  _  lit  -  2 

4  12      ~  3t'  2t  6t 

„   6x  -  1      8x  +  3      4x  -  3  3.7 

(d)~l-       -fflr       -5—'         4i  +  x"15^ 
,    5x  +  1   ,  a;  15x2  —  5x  —  8 


w'2x~3T         2x-3'         3x2  +  6x  +  4 

(?)   x  -  1  =  V3x  -  5,        x2  -  5x  +  6  =  0. 


=  5. 


(0    x  =  2,         x(x  -  1)  =  2(x  -  1). 
0')    2x  =  1,         Sx3  -  12x2  +  6x  =  1. 

2.  Solve  the  following  simultaneous  equations  and  determine  whether 
or  not  the  two  sets  in  each  pair  are  equivalent. 

f    3x  +  2y  =  32,  f  7x  -    y  =  1, 

(a)  \  20x  -  3y  =  1.  t  9x  +  4y  =  70. 

Ans.  (2,  13).  The  two  sets  are  equivalent. 

3x  +  7y  =  2,  f  2x  +  3y  =  0, 

7x  +  8y  =  -  2.  t  4x  +    y  =  -  4. 

2t  =  -  3,  f  s  +  5t  =  3, 


f 
I 


1    i  5s  -  3t  =  -  6.  (.  s  +    t  =  0. 

f  f  *  +    y  =  i,  f  5x  +  4y  =  22, 

1 1*  -  fjr  -  H.  I  3x  +    t/  =  9. 

Ans.  These  two  sets  are  not  equivalent. 

3x  -  2y  =  1,  t  x  -  y  =  1, 

(e)    {  3x  +  4z  =  5,  |  x  +  z  =  1, 

3y  +  5z  =  4.  I  y  +  z  =  0. 
Ans.  (—1,  —  2,  2).          Ans.  An  infinite  number  of  solutions. 


II,  §  21]  REVIEW  OF  EQUATIONS  21 

3.  Eliminate  the  x2  term  from  the  equations 

x2  -  2y*  +  13x  +2y  =  l,  3z2  +  4?/2  -  x  +  Qy  =  3. 

4.  Eliminate  y  from  the  equations 

x2  +  3xy  -  x  +  1  =  0,  2x  +  y  +  1  =  0. 

5.  Eliminate  t  from  the  equations 


, 
l+t*'  1+1?         . 

6.  Eliminate  t  from  the  equations 

J2x  =  <4  +  <2  +  1,  ty  =  P  -  1. 

7.  Eliminate  ra  from  the  equations 

w  =  —  —  mx,  x  =  my. 

m 

8.  Clear  the  following  equations  of  fractions  and  radicals  and 
determine  in  each  whether  the  resulting  equation  is  equivalent  to  the 
given  one  : 


2-3x      3x-l  x  - 


(d) 


4x9x  x2  —  9       a;  +  3 

(e)    x  +  Vx  +  6  =  0.  Cf)    V'6  -  5x  = 


(i) 

9.   How  must  1%  ammonia  and  28%  ammonia  be  mixed  to  get  12 
pints  of  10%  ammonia? 

Ans.  8  and  4  pints. 

10.  Two  given  mixtures  contain  respectively  p%  and  q%  of  a  cer- 
tain ingredient.  Show  that  if  x  units  of  the  first  be  combined  with  y 
units  of  the  second  so  that  the  resulting  mixture  contains  r%  of  this 
ingredient,  then  x:y  =  r  —  p:q  —  r. 


22  MATHEMATICS  [II,  §21 

10.  Assume  that  gravel  has  45%  voids  and  sand  33%,  and  that  4 
bags  of  cement  make  3.8  cu.  ft.,  how  much  cement,  sand,  and  gravel 
are  necessary  to  make  1  cu.  yd.  of  concrete? 

(a)  in  a  1:2:4  mixture.         (fe)  in  a  1  :  3  :  6  mixture. 
(c)   in  a  1  :  2  :  3  mixture.         (d)  in  a  1  :  3  :  5  mixture. 

11.  How  many  pounds  of  skimmilk  must  be  extracted  from  12000 
Ibs.  of  4%  milk  to  raise  the  test  to  4.5%? 

Am.  1333|  Ibs. 

12.  How  many  pounds  each  of  40%  cream  and  skimmilk  are  required 
to  make  125*pounds  of  18%  cream? 

Ans.  56.25  Ibs.  cream,  68.75  Ibs.  skimmilk. 

13.  How  many  pounds  each  of  25%  cream  and  3.5%  milk  are 
required  to  make  130  pounds  of  22.5%  cream? 

Ans.  114.8  Ibs.  of  25%,  15.2  Ibs.  of  3.5%. 

14.  How  much  25%  cream  must  be  added  to  1000  pounds  of  50% 
cream  to  reduce  it  to  40%  cream? 

Ans.  666f  Ibs. 

15.  How  many  pounds  each  of  50%  and  25%  cream  must  be  mixed 
together  to  produce  1000  pounds  of  40%  cream? 

Ans.  600  Ibs.  of  50%,  400  Ibs.  of  25%. 

22.  Polynomials.     Expressions  of  the  form 

1  -  x,  xz  -  3%  +  2,   x+  A/3.T3  +  3.4  +  lxz, 
x*  -  2x*  +  x  -  5 


,  3x«, 


are  examples  of  polynomials  in  x; 


y  -  5  +  4y2,  z5  +  ^  +  A/2z2  -  \ ,  ^  +  a3  -  1  +  2a, 
5  /      o 

are  polynomials  in  y,  z,  and  «,  respectively 

A  polynomial,  in  x  for  example,  is  a  sum  of  terms  each  con- 
taining a  positive  integral  power  of  x  multiplied  by  a  coefficient 
independent  of  x,  and  usually  also  an  absolute  term. 

If  any  number  (value  of  x)  be  substituted  for  x,  the  poly- 
nomial reduces  to  a  number  called  a  value  of  the  polynomial. 


II,  §23]  REVIEW   OF  EQUATIONS  23 

To  each  value  of  x,  which  is  called  the  variable,  there  corre- 
sponds a  unique  value  of  the  polynomial.  For  example,  the 
values  of  x2  —  3x  +  2  which  correspond  to  x  =  0,  x  =  1, 
x  =  \,  are  2,  0,  f . 

The  degree  of  any  term  in  a  polynomial  is  the  exponent  of 
the  variable  in  that  term.  The  degree  of  a  polynomial  is  the 
degree  of  the  term  of  highest  degree  in  it.  Polynomials  are 
usually  arranged  according  to  the  degrees  of  the  terms  and  it  is 
sometimes  convenient  to  supply  with  zero  coefficients  missing 
terms  of  degree  lower  than  the  degree  of  the  polynomial;  thus 

3x2  +  0-x  +  2,         z5  +  0-z4  +  0-z3  +  V2>  +  fz  -  f . 

A  sharp  distinction  is  to  be  made  between  the  coefficients 
and  the  exponents  in  a  polynomial.  The  coefficients  are  very 
general:  they  may  be  any  real  numbers  whatever,  natural 
numbers,  rational  or  irrational  numbers,  positive,  negative,  or 
zero.  On  the  other  hand  the  exponents  are  very  special:  they 
must  be  positive  integers.  Thus  while  the  expressions 

z2  +  1,  ?/  +  y/2,  z2  -  TTZ  +  V2, 
are  polynomials,  the  expressions 

3.1/2  +  1?    yt  _j_   2/y,    z2  -  32  +  Vz, 
are  not. 

23.  Polynomial  of  the  nth  Degree.  A  polynomial  of  de- 
gree n  in  x  (n  being  any  given  natural  number  1,  2,  3,  •  •  •)  can 
be  reduced  by  merely  rearranging  its  terms  and  adding  the 
coefficients  of  like  powers  of  x  to  the  form 


in  which  the  o's  (coefficients)  are  any  real  numbers  (oo  =J=  0 
but  any  or  all  the  others  may  be  zero),  and  the  exponent  n  is  a 
positive  whole  number. 


24  MATHEMATICS  [II,  §24 

24.  Linear  Equations.     An  equation  of  the  first  degree,  or  a 
linear  equation,  in  one  unknown,  x  for  example,  is  the  result 
of  equating  to  zero  a  polynomial  of  the  first  degree  in  x, 

(10)  ax  +  b  =  0         (a  +  0). 

This  equation  has  one  and  only  one  solution.     The  method  of 
finding  the  solution  is  already  known  to  the  student. 

25.  Quadratic  Equations.     An  equation  of  the  second  de- 
gree, or  a  quadratic  equation,  in  x  for  example,  is  the  result  of 
equating  to  zero  a  polynomial  of  the  second  degree  in  x, 

(11)  ax2  +  bx+c  =  Q         (a  4=  0). 

Any  equation  which  can  be  reduced  to  this  form  by  merely 
transposing  and  combining  like  terms  is  also  called  a  quadratic. 
Thus, 

(x  -  l)(x  -  2)  =  Q(x  -  3) 
is  a  quadratic. 

SOLUTION  BY  FACTORING.  If  the  polynomial  axz  +  bx  +  c 
can  be  factored  into  two  linear  factors  in  x  (i.  e.,  polynomials 
of  the  first  degree  in  x)  the  roots  of  the  quadratic  equation 
ax2  +  bx  +  c  =0  can  be  found  by  inspection. 

EXAMPLE  1.  Solve  6x2  +  x  =  15.  Transpose  all  terms  to  the 
left  side  and  factor.  In  order  to  do  this  we  seek  a  pair  of  numbers 
whose  product  is  6  and  another  pair  whose  product  is  —  15  and  such 
that  the  cross  product  is  1.  The  work  may  be  put  down  as  follows: 

6z2  +  x  -  15  =  0 
-5 


This  gives  the  cross  product  13,  but  a  few  trials  of  other  factors  and 
other  arrangements  quickly  leads  to  the  combination 


-3 


II,  §25]  REVIEW  OF  EQUATIONS  25 

which  gives  the  cross  product  1  as  desired.     Hence  the  factors  are 
3x  +  5  and  2x  —  3,  and  we  have  to  solve  the  equation 

(3x  +  5)(2x  -  3)  =  0. 

On  equating  the  first  factor  to  zero  (mentally)  and  solving  we  get 
Xi  =  —  5/3  and  similarly  from  the  second  factor  xz  =  +3/2,  and 
these  are  the  two  solutions  of  the  given  quadratic  equation. 

EXAMPLE  2. 

Q      o  2_  ""  "^ 

3x2  —  i  x  =  • — = — -  . 

Transposing  and  combining  terms  this  reduces  to 

¥  &  +  &  x  =  0 
which  factors  by  inspection  into 

Z(-27Q  X  +  *V)    =  0 

whence  xi  =  0  and  xz  =  —  ?V- 

If  there  are  fractional  coefficients  in  a  quadratic  it  is  usually  best  to 
reduce  it  to  an  equivalent  equation  free  from  fractions  by  multiplying 
every  term  by  the  least  common  multiple  of  all  the  denominators. 
Thus  in  Example  2,  we  could  multiply  every  term  by  21  and  obtain, 

63x2  -  14x  =  3x2  -  15x. 

EXERCISES 

Solve  the  following  quadratic  equations. 

1.  2z2  -  5x  =  3.  2.  10x2  +  x  =  2. 

3.  6x2  +  5x  =  6.  4.  15x2  -  x  =  6. 

5.  6x2  -  5  =  7x.  6.  28x2  -  15  =  x. 

7.  135x2  +  3x  =  28.  8.  78x2  -  x  =  2. 

9.  3?/2  +  y  =  10.  10.  147/2  +  y  =  168. 

11.  Gy2  +  lly  =  35.  12.  15?/2  +  4  =  16y. 

13.  6a2  +  a  =  5.  14.  2a  +  3  =  8a2. 

15.  9a(2a  +  1)  =  14.  16.  10(2a2  -  3)  +  a  =  0. 

17.  3(2s2  -  7)  =  5s.  18.  15(2i2  -  1)  +  7t  =  0. 

19.  p(12p  -  7)  =  10.  20.  5(3r2  -  8)  +  r  =  0. 


26  MATHEMATICS  [II,  §25 

21.  A  z2  +  2x  +  1T47  =  0.  22.  v(y  ~  1}  =  3(y  +  1)  -  2/2. 

23.  2(2  -  1)  =  ;ft(6z  -  1).  24.  P  +  3«  -  1  =  - 


25.  (1  -  e2)z2  -  2px  +  p2  =  0. 

Clear  the  following  equations  of  fractions,  solve  the  resulting  equa- 
tions and  test  their  solutions  in  the  given  equations. 


07 


2x  -  7       x  -  3  '  x2  -  1      2(x  +  1)      4 ' 

3Oi»  1  O  O  A 

M,X i_      zx      —  K  OQ  z        _       d 

z-5z-3  z-1      x-2      x  -  3      x-4' 

26.  Solution  of  a  Quadratic  by  Completing  the  Suqare. 

If  the  polynomial  on  the  left  of  the  quadratic  equation 

ax2  +  bx  +  c  —  0 

cannot  readily  be  factored  by  inspection,  the  equation  can 
be  solved  by  transposing  the  absolute  term  c,  completing  the 
square  of  the  terms  in  x  and  extracting  the  square  roots  of 
both  sides. 

To  complete  the  square  of  ax2  +  bx  is  to  find  a  number  d 
such  that  ax2  +  bx  +  d  is  the  square  of  a  linear  factor  in  x  and 
it  can  always  be  done  as  follows:  1)  extract  the  square  root  of 
the  first  term;  2)  double  this;  3)  divide  this  into  the  second  term; 
4)  square  the  quotient. 

EXAMPLE.     Solve  6x2  —  4x  —  1  =0. 

Transpose  —  1,  and  find  the  number  to  complete  the  square  of 
6x2  -  4x  by  the  above  four  steps:  1)  xV6,  2)  2x>/6,  3)  2/V6,  4)  2/3; 
add  this  to  both  sides: 

6x2  -  4x  +  f  =  f . 

Extracting  the  square  roots,  we  have 

-v-v/fi   —    A/1   =    4-   A/£ 
•*•    'u  »3     —     3I     "?» 

whence  solving  for  x,  we  find 

r,   —  i  -L.  IA/TO  r«  =  l  i  •\/T7) 

•^l    —    3      i^    6    '-*-",  ^2  3    ^^    §    i(XV/. 


II,  §27]  REVIEW   OF  EQUATIONS  27 

The  computations  are  more  easily  made,  if  we  multiply  the  given 
equation  through  by  a  number  which  will  make  the  coefficient  of  x2  a 
perfect  square.  In  the  above  example  we  should  have  to  solve  the 
equivalent  equation, 

36z2  -  24x  +  (        )  =  6. 

The  number  required  to  complete  the  square  is  4, 

36z2  -  24z  +  4  =  10. 
Whence 

6z  -  2  =  ±  VlO 
and 

Si  -i-  |ViO. 


EXERCISES 

Solve  these  equations  by  completing  the  square. 
1.  4z2  +  3z  =  9.  2.  25z2  -  14z  +  1  =  0. 

3.  50z2  +  12x  =  x2  -  i  4.  (x2  +  1)  V3  =  4z. 

5.  12z2  +  5x  =  1.  6.  6^2  +  1  =  6y. 

7.  322  =  13(2  -  1).  8.  x  +  2  =  llar(l  -  x). 

9.  12<2  -  4(o  +  b)t  +  a6  =  0.          10.  2(?/2  +  c2)  =  5ey. 

27.  Solution  of  a  Quadratic  by  a  Formula.  By  the  process 
of  completing  the  square,  a  formula  for  the  roots  of  the  general 
quadratic  equation  can  be  found  as  follows.  Given  the  equation 

(12)  ax*  +  bx  +  c  =  0, 

multiply  through  by  4a,  transpose  4ac,  and  complete  the  square, 

4a2z2  +  4a6x  +  62  =  -  4ac  +  62, 
extracting  the  square  roots,  we  have 


2ax  +  b  =  ±  V62  -  4oc, 
whence  we  find 

-  &  =fc  Vfe2  -  4ac 
~^a^       -• 
which  gives  the  two  roots 


-  6  +      &2  -  4ac  -  6  - 


28  MATHEMATICS  [II,  §27 

This  result  may  be  used  as  a  formula  for  the  solution  of  any 
quadratic  equation  by  substituting  for  a,  6,  c,  of  this  formula 
their  values  from  the  given  equation. 

EXAMPLE.     Solve  3x2  +  4x  —  15  =  0. 
Here  a  =  3,  6  =  4,  c  =  —  15,  and  by  the  formula 


_  -  4  ±  V16  +  180 
6 

whence 

_ -2+7_5 

3/1    "~  « — *  «-.  cLIlCi  3/9    " 

33  a 

EXERCISES 

Solve  the  following  equations  by  the  formula. 

1.  2x2  +  3x  =  4.  2.  x2  =  220  +  9x. 

3.  5x2  +  3x  =  3.  4.  5x2  +  5x  +  1  =  0. 

5.  15?/2  =  86y  +  64.  6.  5z2  =  80  +  21. 

7.  a2  +  a  =  3.  8.  p2  +  3p  =  40. 

9.  I2  +  3a2  =  4o<  -  1.  10.  5m2  +  21m  +  4=0. 

Solve  the  following  equations  by  any  method  and  test  all  results  in 
the  given  equation. 


11.  (2x  -  3)2  =  8x.                            12.  x2  -  2  A 

10         "x        ,    X  ~r  •"         o                             14*C'l 

!3x  +  2  =  0. 

J'  x  +  2  '      2x                                           x 

I  K           X      1                                                 ,  1       *              f) 

x  -  1* 

x(x  -  2)      2x  -  2   '  2x 
ifi        4               !               3              2 

'x  —  1       4  —  x      x  —  2      3— x' 

17.  3x2  +  (9o  -  l)x  -  3a  =  0.        18.  x2  -  2ax  +  a2  -  fc2  =  0. 
19.  c2x2  +  c(a  -  6)x  -  afc  =  0.        20.  x2  -  4ax  +  4a2  -  62  =  0. 

21.  x2  -  6acx  +  a2(9c2  -  462)  =  0. 

22.  (a2  -  62)x2  -  2 (a2  +  62)x  +  a2  -  ft2  =  0. 


II,   §28]  REVIEW  OF  EQUATIONS  29 

Solve  for  y  in  terms  of  x. 

23.   x2  +  12xy  +  9y2  +  3  =  0.         24.   x2  -  4xy  -  4y*  +  x  =  0. 

25.    llx2  +  3Qxy  +  25y*  =  3.  26.    8x2  -  12x?/  +  4y2  =  x  +  1. 

27.    Gx2  -  XT/  -  2y2  =  0.  28.   21x2  =  xy  +  lOy2. 

29.   30x2  +  150*  =  43xy.  30.    12x2  +  41xy  +  35y2  =  0. 

31.  2x2  +  3xy  -  2y2  +  x  +  7y  -  3  =  0. 

32.  3x2  +  lOxy  +  8?/2  +  4x  +  2y  -  15  =  0. 

33.  10x2  +  7xy  +  r/2  -  x  -  2?/  -  3  =  0. 

34.  12x2  =  4xy  +  2ly2  +  2x  +  29y  +  10. 

35.  A  farmer  mows  around  a  meadow  18  X  80  rods.     If  the  swath 
averages  5  ft.  6  in.,  how  many  circuits  will  cut  half  the  grass? 

Ans.    12. 

38.  What  are  eggs  worth  when  2  more  for  a  quarter  lowers  the  price 
5  cents  a  dozen? 

37.  If  the  radius  of  a  circle  be  divided  in  extreme  and  mean  ratio 
the  greater  part  is  the  side  of  the  regular  inscribed  decagon.     What 
is  the  perimeter  of  the  regular  decagon  inscribed  in  a  circle  2  feet  in 
diameter?  Ans.   6.180 

38.  When  a  heavy  body  is  thrown  upward  with  an  initial  velocity 
v  ft.  per  second,  its  distance  from  the  earth's  surface  at  the  end  of  t 
seconds  is  given  by  the  equation  d  —  vt  —  16<2.     If  a  projectile  is  shot 
upward  with  a  muzzle  velocity  of  1000  ft.  per  second,  when  will  it  be 
15,600  ft.  high?  Ans.   30  and  32^  sec. 

28.  Equations  in  Quadratic  Form.  The  terms  of  an  equa- 
tion which  is  not  a  quadratic  in  the  unknown  can  sometimes 
be  grouped  so  as  to  make  it  a  quadratic  in  an  expression  con- 
taining the  unknown.  Thus,  x4  —  13z2  +  36  =  0  is  not  a 
quadratic  in  x  but  it  is  a  quadratic  in  x2]  again  if  the  terms  of 
x*  —  6x3  +  7x2  +  Qx  =  8  be  grouped  in  the  form 


-  2(x2  -  3z)  =  8 
it  is  seen  to  be  a  quadratic  in  (x2  —  3x). 


30  MATHEMATICS  [II,  §28 

EXAMPLE  1.     Solve  6x  -  7Vx  =  20. 

Transpose  20  and  this  can  be  solved  by  the  formula  as  a  quadratic 
in  Vx;  whence, 

r      7  ±  A/49  +  480 

Vx=       "IT" 

and,  since  the  positive  square  root  cannot  be  negative, 

>lx  =  2.5         and        a;  =  6.25. 
We  verify  that  this  satisfies  the  given  equation. 

EXERCISES 

1.x4-  13x2  +  36  =  0. 

2.  x  +  Vx  +  6  =  14.     Ans.  10. 

3.  2x2  +  3  Vx2  -  2x  +  6  =  4x  +  15.     Ans.   -  1  and  3. 

=  2.     Ans.  0  and  1. 


x  +  Vl  -x2 

5.  sVx  +  18  =  5  3Vx~2.     Ans.  8  and  -  729/125. 

6.  x4  -  6X3  +  7x2  +  6x  =  8.    Ans.   -  1,  1,  2,  4. 

29.  Imaginary  Roots.  .  There  are  quadratics  which  are  not 
satisfied  by  any  real  number.  For  example,  z2  =  —  4,  x2  +  2x 
+  2  =  0.  This  is  because  the  square  of  every  real  number 
(except  0)  is  positive.  If  we  attempt  to  solve  the  equation 
x2  +  2x  +  2  =  0  either  by  completing  the  square  or  by  the 
formula  we  are  led  to  the  indicated  square  root  of  a  negative 
number,  and  this  is  not  a  real  number;  thus 


These,  and  other  considerations  have  led  to  the  invention 
of  numbers  whose  squares  are  negative  real  numbers;  they  are 
called  imaginary  numbers.  The  imaginary  unit  is  usually 
denoted  by  i.  Hy  definition,  we  have 


II,  §29]  REVIEW  OF   EQUATIONS  31 

The  number  r-i,  where  r  is  any  real  number  is  called  a  pure 
imaginary  number;  e.  g.,  2i,  5i,  —  3i,  --  %i,  i  V3,  etc.  The 
squares  of  pure  imaginary  numbers  are  negative  real  numbers; 
e.  g.,  (2i)z  =  22i2  =  -  4;  (-  3i)2  =  (-  3)2i2  =  -  9;  (i  VJF)2 
=  -  3. 

Conversely,  the  square  roots  of  negative  real  numbers  are 
imaginary  numbers;  the  square  roots  of  —  4  are  2i  and  —  2i; 
i.  e.,  V^l  =_i  VI"  =  2i,  -  -  V-  4  =  -  i  VI"  =  -  2i;  V^3 

=  i  V3,  —  -V  —  3   =    •      i  V3 ;  in  general,    V—  p  =  i  ^p,  where 
p  is  a  real  positive  number. 

Expressions  of  the  form  2  +  5i,  1  —  i,  3  —  2i,  —  I  +  i, 
etc.,  indicating  the  sum  of  a  real  and  an  imaginary  number  are 
called  complex  numbers.  They  may  be  added,  subtracted, 
multiplied,  and  divided  by  the  laws  of  algebra  as  though  i  were 
a  real  number  and  the  results  simplified  by  putting  —  1  for  i-, 

-  i  for  i3,  +  1  for  i4,  etc. 

We  can  now  say  that  every  quadratic  equation  can  be  solved. 
The  solutions  of  the  equation 

x2  -  2x  +  2  =  0 
may  be  found  by  the  formula, 


2  ±  A/4  -  8 
,--    --^ 

whence 

x\  =  1  +  i        and        xz  =  1  —  i; 

and  we  verify  both  these  answers  as  follows: 

(1  +  i')2  -  2(1  +  i)  +  2  =  0,        (1  -  *)*  -  2(1  -0+2  = 

EXERCISES 

i.  x-  -  4x  +  5  =  0.  2.  xz  +  <ox  +  13  =  0. 

-4ns.  2  ±  i.  Ans.   —  3  ±  2i. 

3.  36x2  -  36x  +  13  =  0.  4.  2x2  +  2x  +  1  =  0. 

Ans.  1/2  ±  t'/3.  -4ns.   —  1/2  ±  t/2. 


32  MATHEMATICS  [II,  §29 

5.  x2  +  4  =  0.  6.  x2  +  x  +  1  =  0. 

Ans.   ±  2i.  Ans.   -  1/2  ±i  V3/2. 

7.  z2  -  2z  +  3  =  0.  8.  x2  -  |x  +  1  =  0. 

9.  x2  -  2x  VJj  +  7  =  0.  10.  2z2  -  2x  +  5  =  0. 

11.  x2  +  3x  +  2.5  =  0.  12.  49z2  -  56x  +  19  =  0. 

30.  The  Sum  and  Product  of  the  Roots.     The   two   roots 
of  the  quadratic  equation 

ax2  +  bx  +  c  =  0 
are  by  (14),  §27, 


-  6  +  V&2  -  4oc  -  6   -  Vb2  -  4oc 

zi  = and          x2  =  — -  . 

2a  2a 

The  sum  of  these  roots  is  —  b/a,  and  their  product  is  +  c/a,  as 
may  be  seen  by  adding  and  multiplying  them  together. 

We  can  thus  find  the  sum  and  the  product  of  the  roots  of  a 
given  quadratic  equation  without  solving  it.  Thus  in  the 
equation, 

36z2  -  36z  +  13  =  0 

the  sum  of  the  roots  is  1,  and  their  product  is  13/36. 
Again  in  the  equation, 

myz  —  4ay  +  4a6  =  0 

the  sum  of  the  roots  is  4a/m,  and  their  product  is  4ab/m. 

31.  Equation  having  Given  Roots.  We  have  seen  that  if 
the  left  member  of  the  quadratic  equation 

ax2  +  bx  +  c  =  0 

can  be  separated  into  linear  factors,  its  roots  can  be  found  by 
inspection.  Therefore  if  we  wish  to  make  up  a  quadratic 
equation  whose  roots  shall  be  two  given  numbers,  r  and  s  for 
example,  we  hare  only  to  write 

a(x  —  r)(x  —  s)  =0 


II,  §33]  REVIEW   OF   EQUATIONS.  33 

and  multiply  out.  The  factor  a  is  arbitrary  and  may  be  chosen 
so  as  to  clear  the  equation  of  fractions  if  desired;  thus,  to  make 
an  equation  whose  roots  shall  be  f  and  —  f ,  we  write, 

a(z-f)(z+f)  =0, 

and  if  we  take  a  =  10,  the  resulting  equation  is 
10z2  -  llz  -  6  =  0. 

32.  Number  of  Roots.     Conversely,  it  is  readily  shown  that 
if  r  and  s  are  roots  of  the  quadratic  equation, 

ox2  +  bx  +  c  =  0 

then  the  left  member  can  be  factored  in  the  form, 
(15)  a(x  -  r)(x  -  s)  =  0 

and  this  shows  that  no  quadratic  can  have  more  than  two 
roots.  Some  quadratics  have  only  one  root;  for  example 
4z2  +  9  =  12x  is  satisfied  only  by  x  =  3/2. 

If  62  —  4ct£  =  0,  then  the  polynomial  ax2  +  bx  +  c  is  a 
perfect  square  and  the  equation  ox2  +  bx  +  c  =  0  has  only 
one  root,  and  conversely. 

For,  if  62  —  4oc  =  0,  then  c  =  62/4a  and  this  is  precisely 
the  number  necessary  to  complete  the  square  of  ox2  +  bx. 
Also  if  62  —  4oc  =  0,  the  formula  (13),  §  27,  gives  not  two  but 
one  root. 

33.  Kind  of  Roots.     If  a,  b,  and  c,  are  real  numbers  and  if 
62  —  4ac  >  0,  then  the  quadratic  equation  ax*  +  bx  +  c  =  0 
has  two  real  roots;  but  if  62  —  4oc  <  0,  the  equation  has  two 
imaginary  roots. 

This  is  seen  at  once  on  noting  the  formula  (13),  §  27,  which 
gives  the  roots. 
4 


34  MATHEMATICS  [II,  §33 

EXAMPLE  1.     4z2  -  12x  +  9  =  0. 

Here  b2  —  4ac  =  144  —  144  =  0,  the  left  member  is  a  perfect  square 
and  the  equation  has  only  one  root. 

EXAMPLE  2.     3x2  -  5x  +  2  =  0. 

Compute  b2  —  4ac  =  25  —  24  =  +  1,  which  shows  that  the  equa- 
tion has  two  real  roots. 

EXAMPLE  3.     x2  +  x  +  I  =  0. 

Here  b2  —  4oc  =  —  3,  which  shows  that  the  equation  has  imaginary 
roots. 

If  a,  b,  and  c,  are  rational  numbers,  then  the  roots  of  the  equation 
ax2  +  bx  +  c  =  0  are  rational  if  b2  —  4ac  is  a  perfect  square,  i.  e.,  the 
square  of  a  rational  number:  in  particular  if  a,  b,  and  c,  are  integers 
and  if  b2  —  4ac  is  a  perfect  square  the  left  member  of  the  equation  can 
be  factored  by  inspection. 

EXAMPLE  4.     2x2  -  x  -  6  =  0. 

Here  b2  —  4ac  =  1  +  48  =  49;  the  left  member  factors  into 

(2x  +  3)(x  -  2)  =  0 
whence  the  roots  are  —  3/2  and  2. 

EXERCISES 

1.  Form  the  equations  whose  roots  are 

(a)  1,  3,  -  5.  (6)  -  2,  3,  -  4,  6. 

(c)  1/3,  -  7/2,  3/5.  (d)  ±  1,  ±  4. 

(e)  ±  V2,  ±  V5.  (/)  0,  -  2,  ±  V^"2. 

(?)  3,  5  ±  >/5.  (ft)  4  ±  A/3,  -  1  ±  VS. 

(i)  -  a,  -  0,  -  7.  0')  ka>  k-P,  ky. 

(k)  <x+k,0^k    y+k.  (/)  I/a,  1//S,  1/7. 

(TO)  a,  ft  7.  (»)  a2,  /S2,  72- 

(0)  a  -  0,  0  -  7,  7  -  «•  (P)  "A  07,  7«- 

2.  Determine  the  nature  of  the  roots  of  the  following  equations. 
(a)  3x2  -  4.r  - 1  =0.  (6)   5x2  +  6z  +  1  =  0. 

(c)  2x2  +  x  -  6  =  0.  (d)  x*  -  2x  -  1  =  0. 

(e)  5x2  -  6x  +  5  =  0.  (/ )  x2  -  6  V3x  -  5  =  0. 

(g)  x2  +  x  +  1  =  0.  (h)   13x2  -  6  >/3x  +  7  =  C. 

(1)  3x2  +  2x  +  1  =  0.  0')    2*2  -  16x  +  9  =  0. 
(*)  5x2  -  12x  -  8  =  0.  (0    6x2  +  4x  -  5  =  0. 


II,  §33]  REVIEW  OF  EQUATIONS  35 

(m)  5x  +  7  =  (3x  +  2)(x  -  1).       (n)  5(x2  +  x  +  1)  =  1  -  16x. 
(o)    2x(x  -  3)  =  7(3x  +  2).  (p)  7(x2  +  5x  +  3)  =  x(l  -  x). 

(9)    3x(x  +  1)  =  (3  -  x)(3  +  x).      (r)    3x2  =  13(x  -  1). 
t(s)     0,  +  2)(V  -  2)  =  2y  -  7.         (0    3fo  +  l)fo  -  1)  =  4y. 
(M)    60(3  -  %)  =  19(0  -  I)2-          (»)    2/2  -  2yV3  +  7  =  0. 

3.  Without  solving  find  the  sum  and  the  product  of  the  roots  of  each 
of  the  equations  in  Ex.  2. 

4.  Determine  the  nature  of  the  roots  of  the  following  equations  in 
which  a,  b,  c  are  known  real  numbers. 

(a)  (x  -  a)2  =  62  +  c2.  (6)    (x  +  a)2  =  862. 

(c)  a  (ax2  +  26x  -  a)  =  b(bx2  -  lax  -  b). 

(d)  7/2  =  2a(y  -b)  +  2b(y  -  a). 

(e)  (a  +  6  -  c)?/2  -  2cy  =  (a  +  b  +  c). 

(/)    (a  +  b  -  c)x2  +  4(o  +  b)x  +  (a  +  b  +  c)  =  0. 

(0)    (b  +  c  -  2a)x2  +  (c  +  a  -  26)x  +  (a  +  6  -  2c)  =0. 

5.  Determine  values  of  a  for  which  each  of  the  following  quadratic 
equations  will  have  equal  roots. 

(a)  x(x  +  4)  +  2a(2x  -  1)  =  0.      (6)    (x  -  I)2  =  2a(3x  -  7)  -  20. 
(c)    (x  -  a)2  =  a2  -  8a  +  15.  (d)   x2  -  15  =  2a(x  -  4). 

(e)   3(x2  +  Sax  -  a)  =  x.  (/)    9x2  +  6(0  -  4)x  +  a2  =  0. 

(g)   3ax(x  -  1)  =  ax  -  2.  (h)    (a  +  l)x2-(a  +  2)x  +fa  =  0. 

(t)    (4a2  +  3)x2  +  8a(3  -  2a)x  +  4(4a2  -  12a  -  3)  =  0. 

6.  Find  a  value  of  k  such  that  the  sum  of  the  roots  of  the  equation 
x2  —  3(k  +  l)x  +  Qk  =  0  shall  be  one  half  their  product. 

7.  Construct  equations  whose  roots  shall  be  greater  by  2  (also  less 
by  2)  than  the  roots  of  the  equations  in  Ex.  2. 

8.  Construct  equations  whose  roots  shall  be  twice  (also  half)  the 
roots  of  the  equations  in  Ex.  2. 


CHAPTER  III 
GRAPHIC  REPRESENTATION 

34.  Graphic  Methods.   The  methods  studied  in  plane  geome- 
try for  constructing  various  figures  when  certain  of  their  dimen- 
sions and  angles  are  known  are  used  extensively  in  making 
designs  for  machines,  plans  for  buildings  and   various  other 
structures,  and  also  for  solving  problems  that  require  the  deter- 
mination of  unknown  dimensions,  angles,  areas,  etc. 

These  methods  often  give  the  desired  results  with  sufficient 
accuracy  for  practical  purposes,  and  they  are  more  direct  and 
rapid  than  numerical  computation.  Of  even  greater  importance 
however  is  their  use  in  checking  the  results  of  calculations,  since 
there  are  always  possibilities  of  error  even  when  great  care  is 
exercised.  It  should  be  emphasized  that  every  practical  calcu- 
lation (i.e.  one  which  is  to  be  used  in  construction,  or  other  ac- 
tual work  where  time,  material,  and  money  will  be  wasted  if 
the  calculation  is  incorrect)  should  always  be  checked  by  some 
independent  means. 

Two  rectilinear  figures  are  similar  if  their  corresponding  angles 
are  equal  and  their  corresponding  dimensions  are  proportional. 

35.  Drawing  to  Scale.     When  two  plane  figures  are  similar, 
each  is  said  to  be  a  scale  drawing  of  the  other.     The  smaller 
is  said  to  be  reduced  or  drawn  to  a  smaller  scale.     For  example, 
if  a  drawing  be  made  of  a  floor  plan  of  a  house  so  that  the  angles 
in  the  drawing  are  equal  to  those  in  the  house  itself,  and  the 
dimensions  of  the  drawing  are  -^  of  those  of  the  house,  it  is  said 
to  be  drawn  to  a  scale  of  ^  inch  to  one  foot.     From  such  a  draw- 
ing the  builder  can  read  off  on  a  scale  divided  into  quarter  inches 
the  dimensions  of  the  parts  he  is  about  to  construct. 

36 


Ill,  §  36]  GRAPHIC  REPRESENTATION  37 

This  method  of  drawing  figures  to  scale,  reading  off  their  un- 
known angles  on  a  protractor,  and  their  unknown  dimensions 
on  a  conveniently  divided  scale,  furnishes  a  graphic  solution  of 
many  problems  and  it  has  many  practical  applications. 

EXAMPLE.  The  distance  AB  =  98  yards, 
Fig.  1,  and  the  angles  PAB  =  51°,  PBA  =  63°, 
having  been  measured  from  one  side  of  a  river, 
the  triangle  can  be  drawn  to  scale  and  the 
width  PR  of  the  river  can  be  read  off  on  the 
scale,  about  75  yards. 

FIG.  1 
EXERCISES 

1.  Find  the  length  of  the  projection  of  the  altitude  of  an  equilateral 
triangle  upon  one  of  its  sides.  Ans.  .75s 

2.  Draw  two  diagonals  through  the  centre  of  a  regular  hexagon. 
Find  the  length  of  the  projection  of  one  of  them  upon  the  other. 

Ans.   s. 

3.  Draw  two  diagonals  through  the  same  vertex  of  a  regular  penta- 
gon.    Find  the  length  of  a  projection  of  one  of  them  upon  the  other. 

Ans.   1.3s 

4.  The  pitch  of  a  roof  is  the  ratio  of  the  height  of  the  ridge   above 
the  plates  to  the  distance  between  the  plates.     Find  the  length  of  the 
rafters  and  their  inclination  for  a  f  pitch  roof  on  a  building  28  ft.  wide. 

Ans.   21.8,  50°. 

5.  Find  the  length  of  the  corner  rafters,  and  also  of  the  middle  rafter 
on  each  side  of  a  square  roof  on  a  house  34  X  34  feet,  the  apex  of  the 
roof  being  12  feet  above  the  top  floor.     Find  also  their  inclinations. 

Ans.   26.9,  20.8,  26°.5,  35°. 

6.  The  roof  of  a  building  36  ft.  wide  is  inclined  at  an  angle  of  54° 
to  the  horizon.     Find  the  length  of  the  rafters,  allowing  2  ft.  overhang. 

Ans.  32.6  ft. 

7.  To  determine  the  horizontal  distance  between  two  points  P  and  Q 
on  the  same  level  but  separated  by  a  hill,  a  point  R  is  selected  from  which 
P  and  Q  are  visible.   Then  PR  =  200  ft.,  QR  =  223  ft.,  and  angle  PRQ  = 
62°  are  measured.     Draw  the  figure  and  scale  off  PQ.  Ans.  210. 


38  MATHEMATICS  [III,  §  36 

8.  The  steps  of  a  stairway  have  a  tread  of  10  in.,  and  a  rise  of  7  in. 
Find  the  inclination  of  the  stringers  to  the  floor.  Ans.   35°. 

9.  Plot  four  points  on  a  sheet  of  paper.    Mark  them  A,  B,  C,  D. 
Construct  a  point  P  one-half  the  way  from  A  to  B,  a  point  Q  one- 
third  the  way  from  P  to  C,  and  a  point  R  one-fourth  the  way  from 
Q  to  D.    Mark  the  four  given  points  in  some  other  order  and  repeat 
the  construction.    What  conclusion  do  you  draw  ? 

36.   Rectangular  Coordinates  of  a  Point  in  a  Plane.     The 
position  of  any  point  in  the  plane  is  uniquely  determined  as 


X'  O 

Y' 

FIG.  2 

soon  as  we  know  its  distance  and  sense  from  each  of  the  two  per- 
pendicular lines  X'  X  and  Y'Y.  These  lines  are  taken  first,  and 
are  drawn  in  any  convenient  position. 

The  distance  from  X'  X  (RP  =  b  in  the  figure)  is  called  the 
ordinate  of  the  point  P.  The  distance  from  Y'Y  (SP  =  a  in 
the  figure)  is  the  abscissa  of  P. 

Abscissas  measured  to  the  right  of  Y'Y  are  positive,  those  to 
the  left  of  Y'Y  are  negative.  Ordinates  measured  above  X' X 
are  positive,  those  below  negative. 

The  abscissa  and  ordinate  taken  together  are  called  the 
coordinates  of  the  point,  and  are  denoted  by  the  symbol  (a,  6). 
In  this  symbol  it  is  agreed  that  the  number  written  first  shall 
stand  for  the  abscissa. 

The  lines  X' X  and  Y'Y  are  called  the  axes  of  coordinates, 
X' X  being  the  axis  of  abscissas  or  the  axis  of  X,  and  Y'Y  the 


HI,   §37] 


GRAPHIC   REPRESENTATION 


39 


axis  of  ordinates  or  the  axis  of  Y;  and  the  point  0  is  called  the 
origin  of  coordinates. 

The  axes  of  coordinates  divide  the  plane  into  four  parts 
called  quadrants.  Figure  3  indicates  the  proper  signs  of  the 
coordinates  in  the  different  quadrants. 


III 


IV 


FIG.  3 


FlG.   4 


37.  Plotting  Points.  To  plot  a  point  is  to  locate  it  with 
reference  to  a  set  of  coordinate  axes.  The  most  convenient 
way  to  do  this  is  to  first  count  off  from  0  along  X'X  a  number 
of  divisions  equal  to  the  abscissa,  to  the  right  or  left  according 
as  the  abscissa  is  positive  or  negative.  Then  from  the  point 
so  determined  count  off  a  number  of  divisions  equal  to  the  ordi- 
nate,  upward  or  downward  according  as  the  ordinate  is  positive 
or  negative.  The  work  of  plotting  is  much  simplified  by  the 
use  of  coordinate  paper,  or  squared  paper,  which  is  made  by  ruling 
off  the  plane  into  equal  squares,  the  sides  being  parallel  to  the 
axes.  Thus,  to  plot  the  point  (4,  —  3),  count  off  four  divisions 
from  0  on  the  axis  of  X  to  the  right,  and  then  three  divisions 
downward  from  the  point  so  determined  on  a  line  parallel  to 
the  axis  of  Y,  as  in  Fig.  4. 

If  we  let  both  x  and  y  take  on  every  possible  pair  of  real 
values,  we  have  a  point  of  the  plane  corresponding  to  each  pair 
of  values  of  (x,  ?/).  Conversely,  to  every  point  of  the  plane 
corresponds  a  pair  of  values  of  (x,  y). 


40 


MATHEMATICS 


[HI,  §37 


EXERCISES 

1.  Plot  the  following  points   (3,  3),    (4,  5),   (-  2,  3),   (-  4,  -  2), 
(7,  -  2),  (0,  4),  (0,  -  4),  (3,  0),  (-  3,  0),  (0,  0). 

2.  What  is  the  y-coordinate  of  any  point  on  the  x-axis? 

3.  What  is  the  z-coordinate  of  any  point  on  the  y-axis? 

4.  Show  that  the  line  joining  (5,  4)  and  (—5,  —  4)  is  bisected  by 
the  origin. 

5.  Find  the  distance  from  the  origin  to  each  of  the  points  in  Ex.  1. 

6.  Find  the  lengths  of  the  sides  of  the  triangle  whose  vertices  are 
(1,  1),  (5,  2),  (3,  4).  Ans'.   Vl7;   Vl3;  2V2. 

7.  What  is  the  abscissa  of  any  point  upon  a  straight  line  parallel 
to  the  y-axis  and  four  units  to  its  right? 

8.  What  is  the  ordinate  of  any  point  upon  a  straight  line  parallel 
to  the  re-axis  and  three  units  above  it? 

9.  (a)  What  relation  exists  between  the  coordinates  of  any  point 
of  a  line  bisecting  the  angle  between  the  positive  directions  of  the 
two  axes?     (6)  Between  the  positive  direction  of  the  y-axis  and  the 
negative  direction  of  the  x-axis? 

10.  What   relation  would   exist   between   the   coordinates   of   any 
point  of  the  line  in  Ex.  9  (a),  if  it  were  raised  four  units  parallel  to 
itself?     If  it  were  lowered  five  units? 

38.  Statistical  Data.  The  following  table  shows  the  rainfall 
in  inches,  as  observed  at  the  Agricultural  Experiment  Station  at 
LaFayette,  Indiana,  by  months  for  1916,  1917,  and  the  average 
for  the  past  30  years. 


Jan. 

Feb. 

Mar. 

i~08 

Apr. 

May 

Jun. 

Jul. 

Aug. 

Sep. 

Oct. 

2~25 

Nov. 

Dec. 

1916     .     . 

740 

1  16 

1  57 

582 

527 

3  56 

1  81 

222 

225 

4.79 

1917     .     . 

1  54 

1  25 

409 

432 

475 

541 

1  47 

409 

1  03 

522 

0  13 

0.68 

Average    . 

3.11 

2.88 

3.78 

3.38 

4.05 

3.75 

3.54 

3.32 

3.03 

2.46 

3.23 

2.71 

While  it  is  possible  by  a  study  of  this  table  to  compare  the 
rainfall  month  by  month  in  the  same  year,  or  for  the  same  month 
in  the  two  years,  or  any  month  with  the  normal  for  that  month, 


Ill,  §  38] 


GRAPHIC  REPRESENTATION 


41 


these  comparisons  are  more  easily  made  and  the  facts  are  pre- 
sented much  more  emphatically  by  the  diagram  shown  in  Fig.  5. 
This  is  made  from  the  data  of  the  table  as  follows.  The  24 
vertical  lines  represent  the  months  of  the  two-year  period.  The 
altitudes  of  the  horizontal  lines  represent  inches  of  rainfall.  The 
height  (ordinate)  of  the  point  marked  on  any  vertical  line  shows 
the  rainfall  for  that  month.  The  points  are  connected  by  lines 
to  aid  the  eye  in  following  the  march  of  the  rainfall.  The  full 
line  represents  the  rainfall  for  1916  and  1917,  the  dotted  line 
the  normal  rainfall  as  shown  by  the  experience  of  30  years. 


J   F  M  A   M  J    JASONDJFMAMJJA    S    O  N  D 

MONTHS    1910  .MONTHS    1817 

FIG.  5 

Rainfall  is  a  discontinuous  phenomenon.  Moisture  is  not  pre- 
cipitated continuously,  but  intermittently.  However,  if  we 
make  a  similar  diagram  showing  the  temperature  at  each  hour 
of  the  day  we  might  have  inserted  many  other  points.  We 


42 


MATHEMATICS 


[HI,  §  38 


think  of  the  change  in  temperature  as  a  continuous  phenome- 
non ;  e.g.  when  the  temperature  rises  from  42°  at  8  A.M.  to  51° 
at  9  A.M.,  we  think  of  it  as  having  passed  through  every  inter- 
vening degree  in  that  hour.  Thus  we  can  think  of  the  points 
which  represent  the  temperature  on  the  diagram  from  instant 
to  instant  as  lying  thick  on  a  continuous  curved  line.  This 
curve  is  called  the  temperature  curve. 

In  making  a  graph  of  a  discontinuous  function  like  rainfall, 
we  connect  the  points  with  straight  lines  as  in  Fig.  5,  but  in  case 
of  a  continuous  function  like  temperature,  a  smooth  curve  which 
passes  through  all  the  plotted  points  is  the  best  graphic  repre- 
sentation of  the  function. 

EXERCISES 

1.    Make  a  temperature  graph  from  the  following  data, 


Hour,  A.  M. 
Temperature 

12 
45 

1 
45 

2 
45 

3 
45 

4 
43 

5 

42 

6 
41 

7 
40 

8 

42 

9 
51 

10 

57 

11 
59 

12 
62 

Hour,  p.  M. 
Temperature 

1 

66 

2 

70 

3 

74 

4 
76 

5 
76 

6 
75 

7 
74 

8 
73 

9 

72 

10 
70 

11 
69 

12 

68 

2.  Determine  from  Fig.  5  which  were  the  dry  months  in  1916.     In 
1917.     To  what  extent  do  they  agree  with  each  other  and  with  the 
normal  ? 

3.  Do  as  directed  in  Ex.  2  for  the  wet  months. 

4.  What  straight  line  in  Fig.  5  would  represent  the  average  monthly 
rainfall  for  1916?     For  1917?     For  the  past  30  years? 

5.  To  what  extent  does  the  dotted  line  in  Fig.  5  enable  you  to  pre- 
dict the  probable  rainfall  in  any  given  month  subsequent  to  1917? 

6.  Procure  the  census  data  and  plot  the  population  graph  of  the 
United  States  by  decades  for  a  century. 

7.  Plot  a  graph  of  the  attendance  of  students  at  your  college  or  Uni- 
versity for  as  many  years  back  as  you  can  secure  the  data. 

8.  The  following  data  give  the  Chicago  price  per  bu.  of  No.  2  corn 
by  months  from  Jan.,  1903,  to  May,  1908.     Plot  the  data  using  years 
as  abscissas  and  price  as  ordinates. 


Ill,   §38] 


GRAPHIC   REPRESENTATION 


43 


Jan. 

Feb. 

Men. 

Apr. 

May. 

June. 

July. 

Aug. 

Sept. 

Oct. 

Nov. 

Dec. 

1903 
1904 
1905 
1906 
1907 
1908 

43 
42 
41 
42 
39 
59 

42 
46 
42 
41 
43 
56 

41 

49 
45 
39 

43 

58 

41 

46 
46 
43 
44 
65 

44 
47 
48 
47 
49 
70 

47 
53 
51 
50 
51 

49 
47 
53 
49 
52 

50 
51 
53 

48 
54 

45 
51 
51 
47 
60 

43 
50 
50 
44 
55 

41 
50 
45 

44 
55 

41 

43 
42 
40 

57 

9.  Find  from  the  graph  that  month  in  each  year  in  which  the  highest 
price  occurred.     The  lowest  price.     Find  the  difference  for  each  year 
between  the  highest  and  lowest  price  for  that  year.     Does  there  appear 
to  be  any  relation  between  these  prices  and  the  period  of  harvest? 

10.  The  following  data  gives  the  Chicago  price  of  No.  2  oats  by 
months  from  Jan.,  1903  to  May,  1908.     Plot  the  data  using  years  as 
abscissas  and  price  as  ordinates. 


Jan. 

Feb. 

Mch. 

Apr. 

May. 

June. 

July. 

Aug. 

Sept. 

Oct. 

Nov. 

Dec. 

1903 
1904 
1905 
1906 
1907 
1908 

31 
36 
29 
31 
33| 
48 

33 
39 
29 
29 
37 
48 

31 
38 
29 
28 
39 
52 

32 
36 
28 
30 
41 
51 

33 
39 
28 
32 
44 
53 

35 
39 
30 
33 
41 

33 

38 
27 
30 
41 

33 
31 
25 
29 
44 

35 
29 
25 
30 
51 

34 

28 
27 
32 
45 

33 
29 
29 
33 

44 

34 

28 
29 
33 
46 

11.  Handle  the  data  in  Ex.  10  as  directed  in  Ex.  9. 

12.  A  restaurant  keeper  finds  that  if  he  has  G  guests  a  day  his  total 
daily  expenditure  is  E  dollars  and  his  total  daily  receipts  are  R  dollars. 
The  following  numbers  are  averages  obtained  from  the  books: 


G.. 

210 

270 

320 

360 

E  

16.70 

19.40 

21.60 

23.40 

R  

15.80 

21.20 

26.40 

29.80 

Plot  two  curves  on  the  same  set  of  axes  in  each  case  using  G  as  abscissas. 
For  one  curve  use  E  as  ordinates,  for  the  other  use  R  as  ordinates. 

Below  what  value  of  G  does  the  business  cease  to  be  profitable? 
Connect  the  points  (G,  E)  by  a  smooth  curve.  Continue  this  curve 
until  it  cuts  the  line  (7  =  0.  What  is  the  meaning  of  the  ordinate  E 
for  G  =  0?  Through  what  point  ought  the  curve  connecting  the 
points  (G,  R)  to  pass?  Ans.  (0,  0). 


44 


MATHEMATICS 


[III,  §38 


13.  The  population  of  the  United  States  by  decades  was  as  follows. 
Plot,  and  estimate  the  population  for  1920. 


Year. 

Population. 

Year. 

Population. 

Year. 

Population. 

1790.  . 
1800.... 
1810.... 
1820  

3,929,214 
5,308,433 
7,229,881 
9,663,822 

1840...  . 
1850.  .  .  . 

I860.... 
1870  

17,069,453 
23,191,876 
31,443,321 
38,558,371 

1890.. 
1900.  .  .  . 
1910..  .  . 

62,669,756 
76,295,200 
91,972,266 

1830  

12,806,020 

1880  

50,155,783 

14.  The  football  accidents  for  the  years  given  are  as  follows: 


Year. 

Deaths. 

Injuries. 

Year. 

Deaths. 

Injuries. 

1901.... 
1902.... 

1903.... 
1904.... 
1905.  .  .  . 
1906 

7 
15 
14 
14 
24 
14 

74 
106 
63 
276 
200 
160 

1907... 

1908.... 
1909.... 
1910.... 
1911.... 

15 
11 
30 
22 
11 

166 
304 
216 
499 
178 

Plot  two  curves,  using  the  years  as  abscissas  and  the  deaths  and  injuries 
respectively  as  ordinates. 

15.  The  monthly  wages  in  dollars  of  a  man  for  each  of  his  first  13 
years  of  work  was  as  follows:  28,  30,  37.50,  45,  60,  65,  90,  95,  95,  137, 
162,    190,  210.     Plot  the  curve  showing  the  change.     Estimate  his 
salary  for  the  fourteenth  and  fifteenth  years.     Can  you  be  certain  of 
his  salaries  for  these  years? 

16.  Of  100,000  persons  born  alive  at  the  same  time  the  following 
table  shows  the  number  dying  in  the  respective  age  intervals : 


Months. 

Deaths. 

Months. 

Deaths. 

0-1 

4,377 

6-  7 

579 

1-2 

1,131 

7-  8 

533 

2-3 

943 

8-  9 

492 

3-4 

801 

9-10 

456 

4-5 

705 

10-11 

421 

5-6 

635 

11-12 

389 

Ill,  §38] 


GRAPHIC   REPRESENTATION 


45 


Years. 

Deaths. 

Years. 

Deaths. 

0-   1 

11,462 

19-  20 

344 

1-  2 

2,446 

29-  30 

479 

2-  3 

1,062 

39-  40 

644 

3-  4 

666 

49-  50 

873 

4-  5 

477 

59-  60 

1,404 

5-  6 

390 

69-  70 

,  1,974 

6-  7 

327 

79-  80 

1,854 

7-  8 

274 

89-  90 

571 

8-  9 

234 

99-100 

25 

9-10 

204 

106-107 

1 

Plot  the  above  data.  Make  two  graphs.  In  each  graph  use  deaths 
as  ordinates;  in  one  use  months  as  abscissas,  in  the  other  use  years. 
When  is  the  ordinate  smallest?  largest?  Does  a  small  ordinate  for  the 
years  99-100  and  106-107  indicate  a  low  death  rate?  Explain.  Note 
the  continuous  decrease  in  the  ordinate  of  the  first  curve. 

17.  Using  the  data  below  and  on  p.  46,  plot  a  curve  using  years 
as  abscissas  and  price  of  corn  as  ordinates.     Do  you  notice  any  reg- 
ularity in  the  number  of  years  elapsing  between  successive  high  prices? 
successive  low  prices?     Draw  like  graphs  for  the  other  crops  listed? 

18.  Plot  the  prices  for  the  yrs.  74,.  81,  87,  90,  94,  01,  08,  11,  1916. 
What  do  you  observe  from  this  curve  as  to  the  tendency  in  the  high 
price  of  corn?     Do  you  observe  any  tendency  in  the  lowest  prices  of 
corn  that  is  in  the  prices  for  the  yrs.  72,  78,  84,  89,  96,  02,  06,  1910? 

AVERAGE  FARM  PRICE  DECEMBER  FIRST 
Data  from  the  year  book  of  the  Department  of  Agriculture  1916 


Year. 

Corn. 

Wheat. 

Oats. 

Barley. 

Rye. 

Potatoes. 

Hay,  $  per 
Ton. 

1870. 

49.4 

94.4 

39.0 

79.1 

73.2 

65.0 

12.47 

1871. 

43.4 

114.5 

36.2 

75.8 

71.1 

53.9 

14.30 

1872. 

35.3 

111.4 

29.9 

68.6 

67.6 

53.5 

12.94 

1873. 

44.2 

106.9 

34.6 

86.7 

70.3 

65.2 

12.53 

1874. 

58.4 

86.3 

47.1 

86.0 

77.4 

61.5 

11.94 

1875. 

36.7 

89.5 

32.0 

74.1 

67.1 

34.4 

10.78 

1876. 

34.0 

97.0 

32.4 

63.0 

61.4 

61.9 

8.97 

1877. 

34.8 

105.7 

28.4 

62.5 

57.6 

43.7 

8.37. 

1878. 

31.7 

77.6 

24.6 

57.9 

52.5 

58.7 

7.20 

1879. 

37.5 

110.8 

33.1 

58.9 

65.6 

43.6 

9.32 

Continued  on  p.  46. 


46 


MATHEMATICS 


[HI,   §38 


AVERAGE  FARM  PRICE,  DECEMBER  FIRST 

Continued. 


Year. 

Corn. 

Wheat. 

Oats. 

Barley. 

Rye. 

Potatoes. 

Hay,  S  per 
Ton. 

1880. 

39.6 

95.1 

36.0 

66.6 

75.6 

48.3 

11.65 

1881. 

63.6 

119.2 

46.4 

82.3 

93.3 

91.0 

11.82 

1882. 

48.5 

88.4 

37.5 

62.9 

61.5 

55.7 

9.73 

1883. 

42.4 

91.1 

32.7 

58.7 

58.1 

42.2 

8.19 

1884. 

35.7 

64.5 

27.7 

48.7 

51.9 

39.6 

8.17 

1885. 

32.8 

77.1 

28.5 

56.3 

57.9 

44.7 

8.71 

1886. 

36.6 

68.7 

29.8 

53.6 

53.8 

46.7 

8.46 

1887. 

44.4 

68.1 

30.4 

51  9 

54.5 

68.2 

9.97 

1S88. 

34.1 

92.6 

27.8 

59.0 

58.8 

40.2 

8.76 

1889. 

28.3 

69.8 

22.9 

41.6 

42.3 

35.4 

7.04 

1890. 

50.6 

83.8 

42.4 

62.7 

62.9 

75.8 

7.87 

1891. 

40.6 

83.9 

31.5 

52.4 

77.4 

35.8 

8.12 

1892. 

394 

62.4 

31.7 

47.5 

54.2 

66.1 

8.20 

1893. 

36.5 

53.8 

29.4 

41.1 

51.3 

59.4 

8.68 

1894. 

45.7 

49.1 

32.4 

44.2 

50.1 

53.6 

8.54 

1895. 

25.3 

50.9 

19.9 

33.7 

44.0 

26.6 

8.35 

1896. 

21.5 

72.6 

18.7 

32.3 

40.9 

28.6 

6.55 

1897. 

26.3 

80.8 

21.2 

37.7 

44.7 

54.7 

6.62 

1898. 

28.7 

58.2 

25.5 

41.3 

46.3 

41.4 

6.00 

1899. 

30.3 

58.4 

24.9 

40.3 

51.0 

39.0 

7.27 

1900. 

35.7 

61.9 

25.8 

40.9 

51.2 

43.1 

8.89 

1901. 

60.5 

62.4 

39.9 

45.2 

55.7 

76.7 

10.01 

1902. 

40.3 

63.0 

30.7 

45.9 

50.8 

47.1 

9.06 

1903. 

42.5 

69.5 

34.1 

45.6 

54.5 

61.4 

9.07 

1904. 

44.1 

92.4 

31.3 

42.0 

68.8 

45.3 

8.72 

1905. 

41.2 

74.8 

29.1 

40.5 

61.1 

61.7 

8.52 

1906. 

39.9 

66.7 

31.7 

41.5 

58.9 

51.1 

10.37 

1907. 

51.6 

87.4 

44.3 

66.6 

73.1 

61.8 

11.68 

1908. 

60.6 

92.8 

47.2 

55.4 

73.6 

70.6 

8.98 

1909. 

57.9 

98.6 

40.2 

54.0 

71.8 

54.1 

10.50 

1910. 

48.0 

88.3 

34.4 

57.8 

71.5 

55.7 

12.14 

1911. 

61.8 

87.4 

45.0 

86.9 

83.2 

79.9 

14.29 

1912. 

48.7 

76.0 

31.9 

50.5 

66.3 

50.5 

11.79 

1913. 

69.1 

79.9 

39.2 

53.7 

63.4 

68.7 

12.43 

1914. 

64.4 

98.6 

43.8 

54.3 

86.5 

48.9 

11.12 

1915. 

57.5 

92 

36.1 

51.7 

839 

61.6 

10.70 

1916. 

88.9 

160.3 

52.4 

88.2 

122.1 

146.1 

10.59 

Ill,   §40] 


GRAPHIC   REPRESENTATION 


47 


39.  Other  Graphic  Methods.     The  statistical  data  given  in 
the  preceding  articles  has  been  studied  by  means  of  curves  or 
graphs  drawn  on  rectangular  cross-section  paper.     There  arc 
other  important  methods  of  representing  statistical  data.     Of 
these  methods  we  will  give  names  to  three: 

(1)  Bar  diagrams  or  columnar  charts. 

(2)  Dot  diagrams. 

(3)  Circular  diagrams. 

These  methods  are  best  explained  by  means  of  examples. 

40.  Bar  Diagrams.     Below  is  given  a  bar  diagram  or  chart 
comparing  the  average  size  of  farms  for  the  years  1900  and 
1910  for  the  states  indicated. 

Sizes  of  Farmsdn  Hundreds  of  Ac-res 


3                       6                       9                     12                     15 

W  (joining 
California 
Arizona 
Nebraska 
Missouri 
Michigan 
Georgia 
Alabama 
New  York 
Delaware 

Legend: 
mm&lO 

c=l  1900 

. 

FIG.  6 

EXERCISES 

Make  a  bar  diagram  from  the  following  data: 

1.  The  number  of  cattle  in  millions  on  farms  for  1900  and  1910  in 
the  following  states  were  as  follows. 


Date. 

Texas. 

Iowa. 

Nebraska. 

New  York. 

Oklahoma. 

Indiana. 

1910... 

7.0 

4.5 

2.9 

2.4 

2.0 

1.3 

1900.  .. 

10.0 

5.5 

3.2 

2.6 

3.3 

1.7 

48  MATHEMATICS  [III,  §40 

2.  The  sheep  on  farms  in  millions  in  1910  and  1900  were  as  follows. 


Date. 

Texas. 

Iowa. 

Nebraska. 

New  York. 

Oklahoma. 

Indiana. 

1910... 

1.6 

1.1 

0.3 

0.9 

0.1 

1.3 

1900.  .  . 

1.8 

1.0 

0.5 

1.7 

0.15 

1.7 

3.  Make  a  bar  diagram  comparing  the  number  of  hours  work  re- 
quired by  hand  and  machine  labor  in  producing  selected  units  (U.  S. 
labor  bulletin  54). 


Description  of  Unit. 

Number  of  Hours  Worked. 

Hand. 

Machiue. 

Corn  50  bu.  husked.     Stalk  left  

48.44 
223.78 
284.00 
160.63 
247.54 
125.00 
137.50 
115.28 
171.05 

18.91 
78.70 
92.63 
7.43 
86.36 
12.50 
28.33 
80.67 
94.30 

Seed  1000  Ibs.  cotton  

Harvesting  and  baling  8  tons  timothy.  .  .  . 
Wheat  50  bu  

Potatoes  500  bu  

Butter  500  Ibs.  in  tubs  

5000  cotton  flour  sacks  

Quarry  100  tons  limestone  

Mine  50  tons  bituminous  coal  

4.  Make  a  bar  diagram  comparing  the  value  of  farm  property  for 
the  two  years  1900  and  1910. 


Year  

1910. 

1900. 

Land  

28,475,674,169 

13,058,007,995 

Buildings  

6,325,451,528 

3,556,639,496 

Implements  and  machinery  

1,265,149,783 

749,775,970 

Domestic  animals,  poultry,  and  bees 

4,925,173,610 

3,075,477,703 

5.  Make  a  bar  diagram  of  the  population  of  the  following  states 
for  the  years  1900  and  1910. 


Date. 

Colorado. 

Nevada. 

Idaho. 

Washington. 

Oregon. 

California. 

1910  
1900  

799,024 
539,700 

81,875 
42,335 

325,594 
161,772 

1.141,990 
518,103 

672,765 
415,536 

2,377,549 
1,485,053 

HI,   §41] 


GRAPHIC  REPRESENTATION 


41.  Double  Bar  Diagrams.  In  certain  diagrams  it  is  ad- 
vantageous to  have  the  bars  extend  in  both  directions  from  the 
base  line  as  in  the  following  figure  which  gives  the  distribution 
by  age  and  sex  of  the  total  population  for  1910. 


Males 


Females 


'12    10     8     0     4     2      0      2     4      6     8    10  12 
Hundreds  of  Thousands 

FlG.  7 

EXERCISES 

Make  corresponding  figures  for  the  distribution  by  age  periods  and 
sex  for  1910,  in  per  cents,  of 


1.   Native  Whites  of  Native  Parentage. 

2.  Negroes. 

Age. 

Male. 

Female. 

Male. 

Female. 

Under  5  

6.7 

6.0 
5.5 
5.2 
4.7 
4.1 
3.5 
3.2 
2.6 
2.2 
2.1 
1.6 
1.3 
1.0 
0.6 

6.5 
5.8 
5.3 
5.1 
4.7 
4.0 
3.4 
3.0 
2.4 
2.0 
1.8 
1.4 
1.2 
0.9 
0.6 

6.4 
6.3 
5.9 

5.2 
4.9 
4.3 
3.4 
3.3 
2.3 
2.0 
1.8 
1.2 
1.0 
0.7 
0.4 

6.5 
6.4 
5.9 
5.6 
5.6 
4.7 
3.4 
3.2 
2.3 
1.9 
1.5 
1.0 
0.9 
0.6 
0.4 

5-9    

10-14        

15-19  

20-24  

25-29  

30-34  

35-39  

40-44    

45-49    

50-54  

55-59  

60-64  

65-69  

70-74      .      ... 

50 


MATHEMATICS 


[HI,  §41 


3.  Make  a  diagram  displaying  the  following  data  on  the  average 
yields  and  values  per  acre  of  Iowa  farm  crops  for  1909. 


Crop. 

Yield. 

Value. 

Corn  

37.1  bu. 

$18.60 

Oats  

27.5 

10.54 

Wheat  

15.3 

14.62 

Barley                                

19.2 

9.31 

Rve  

13.6 

8.50 

Flaxseed  

9.1 

11.74 

Timothy  seed      

4.2 

5.79 

Hay  and  forage      

32.0  cwt. 

11.76 

Potatoes  .  .                         

86.8  bu. 

39.10 

4.  Make  a  diagram  showing  the  weight  in  pounds  and  value  of  the 
dairy  products  shipped  from  Humboldt  County,  California,  in  1913. 


Article. 

Weight  in  Lbs. 

Value. 

Butter                .            

5,793,620 

$1,796,190 

Cheese  

304,570 

54,820 

Condensed  milk  

1,302,560 

112,720 

Dry  milk  

1,692,100 

157,430 

Fresh  cream  and  buttermilk  

277,800 

6,920 

Casein  ....        

1,484,910 

89,100 

42.  Dot  Diagrams.  The  following  diagram  taken  from  the 
U.  S.  census  reports  gives  the  number  of  all  sheep  on  farms 
April  15,  1910. 


JV.  Dak.  ^ 
•      Q    ' 


S.  Dak. 


Arcb. 
•     0 


LEGEND 
•  200,000 

9  150,000  to  200,000 
3  100,000  to  150,000 
Q  50,000  to  100,000 
O  less  than  50,000 


FIG.  8 


HI,  §43] 


GRAPHIC   REPRESENTATION 


51 


EXERCISES 

1.  Make  a  corresponding  chart  showing  all  sheep  on  farms  April  15, 
1910  for 


Wyoming  

5,397,000 

Utah  

1,827,000 

Montana   

5,380,000 

Colorado  

1,400,000 

Idaho                

3,010,000 

Nevada  

1,150,000 

2.  Make  a  dot  diagram  showing  all  fowls  on  farms  in  the  states 
given  on  April  15,  1910.  [Here  it  is  convenient  to  let  •  stand  for 
1,000,000.] 


North  Dakota 

3,268,000 

Iowa            

23,482,000 

South  Dakota  

5,251,000 

Minnesota  

10,697,000 

Nebraska     

9,351,000 

Montana  

966,000 

43.  Circular  Diagrams.  The  following  diagram  shows  the 
relative  percentage  of  improved  and  unimproved  land  area  in 
farms  for  the  total  land  area  of  the  U.  S.  1850-1880-1910. 
(U.  S.  census  report  1910.) 


1850 


1910 


FIG.  9 


FIG.  10 


FIG.  11 


The  circles  indicate  by  the  size  of  their  sectors  the  relative 
ratio  of  lands  improved  and  unimproved  in  farms  to  the 
total  land  area  of  the  U.  S.  Note  the  rapid  decrease  in  the 
area  not  in  farms,  also  the  increase  in  the  proportion  of  improved 
to  the  unimproved.  In  1910  less  than  50%  of  the  total  area  is 
in  farms. 


52 


MATHEMATICS 


[HI,  §43 


EXERCISES 

1.  Make  a  circular  diagram  showing  in  percents  the  relative  im- 
portance of  the  several  countries  in  the  production  and  consumption  of 
cotton. 


United  States 60.9 

India 17.1 

Egypt 6.6 

China .  .  .5.4 


Russia 4.5 

Brazil 1.9 

All  others..  ..3.6 


2.  Make  circular  diagrams  showing  per  cent,  distribution  of  foreign 
born  population  by  principal  countries  of  birth  for  the  years  indicated. 


1850. 

1870. 

1890. 

1910. 

Germany  

26.0 

30.4 

30.1 

18.5 

Ireland  

42.8 

33.3 

20.2 

10.0 

Canada  and  New  Foundland  

6.6 

8.9 

10.6 

9.0 

Great  Britain  

16.9 

13.8 

13.5 

9.0 

Norway,  Sweden  and  Denmark  

0.8 

4.3 

10.1 

9.3 

Austria-Hungary  

1.3 

3.3 

12.4 

Russia  and  Finland  

0.1 

0.1 

2.0 

12.8 

Italy  

0.2 

0.3 

2.0 

9.9 

All  others  

6.6 

7.6 

8.2 

9.1 

3.  Make  circular  diagrams  for  the  years  1900  and  1910  showing  per 
cent,  of  total  value  of  farm  property  represented  by  the  items  men- 
tioned. 


1910. 

1900. 

Land         

69.5 

63.9 

Buildings 

15.4 

17.4 

Implements  and  machinery  T  . 

3.1 

3.7 

Domestic  animals,  poultry,  and  bees 

12.0 

15.0 

(Compare  with  Ex.  4,  p.  48.) 

44.  Different  Shadings  or  Colors  are  sometimes  used  in 
maps  to  represent  different  statistical  facts.  The  annexed  chart 
gives  the  average  value  of  farm  land  per  acre  in  Delaware. 
The  average  for  the  state  is  $33.63. 


Ill,   §45] 


GRAPHIC   REPRESENTATION 


53 


Legend. 

H  $10  to  $25    per  acre 

'3  $50  to  $75    per  acre 

[^  HJ  $75  to  $100  per  acre 

FIG.  12 


EXERCISES 

1.  Draw  a  map  of  Connecticut  showing  the  counties  and  mark  to 
show  the  average  value  of  farm  land  per  acre.     Average  value  for  state 
is  $33.03.     Average  value  by  counties  is:    Fairfield  $75  to  $100  per 
acre.     New  Haven  and  Hartford    $25  to   $50  per  acre.     Litchfield, 
Tolland,  Windharn,  Middlesex,  and  New  London  $10  to  $25  per  acre. 

2.  Draw  a  map,  give  legend,  and  mark  to  show  per  cent,  of  im- 
proved land  in  farms  operated  by  tenants  by  states  in  1910.     Utah, 
less  than  10  per  cent.     Wyoming,  10  to  20  per  cent.     Colorado  and 
Missouri,  20  to  30  per  cent.     Kansas,  Nebraska,  and  Iowa,  30  to  40 
per  cent.     Illinois,  40  to  50  per  cent. 

45 .  Distance  between  two  Points.  Let  PI  and  P2  be  the  end 
points  of  a  given  segment  in  the  plane.  PI  and  P2  are  given 
points,  i.e.,  their  coordinates  (Xi,  YI)  and  (^2,  ^2)  are  given 
or  known  numbers. 

We  wish  to  find  the  length 
of  the  segment  PiP2  in  terms 
of  xi,  i/i,  xz,  yz ;  or,  in  other 
words  to  find  the  distance 
between  two  given  points. 

Through  PI  draw  a  line 
parallel  to  the  a>axis,  and 


M, 


FIG.  13 
through  PZ  a  line  parallel  to  the  i/-axis  intersecting  the  first 


54  MATHEMATICS  [III,   §46 

in  S.  Then  whatever  the  relative  positions  of  PI  and  P%  in  the 
plane,  the  measure  of  PiS  is  x*  —  x\,  and  the  measure  of  SP2  is 
7/2  -  y\ ;  also 

pj%  =  !\s*  +  isp?. 

Therefore  if  we  let  d  represent  the  required  distance  PiP2, 


(1)  d  =      (x2  -  xtf  +  (yz  -  7/02. 

It  is  clearly  immaterial  which  of  the  two  points  is  called  PI 
and  which  P2,  so  the  formula  may  also  be  written  in  the  equiva- 
lent form 


(1)  d  =  V(Xl  -  x2Y  +  (y,  -  ytf, 

and  may  be  expressed  in  words  thus  :    The  distance  between  two 
points  given  by  their  rectangular  coordinates  is  equal  to  the  square 
root  of  the  sum  of  the  square  of  the  difference  of  the  abscissas  and 
.  the  square  of  the  difference  of  the  ordinates. 

EXAMPLE.     The  distance  from  the  point  (2,  —  7)  to  the  point  (7,  5)  is 


d  =     52  +  122  =  13. 

46.  Ratio  of  Division.  Let  PI  and  P2  be  two  fixed  points 
on  a  line  and  P  any  third  point.  Then  the  point  P  is  said  to 
divide  the  segment  PiP2  in  the  ratio 


This  ratio  X  is  called  the  ratio  of  division  or  the  division  ratio. 


PI  P  PI 

FIG.  14 


If  we  choose  an  origin  0  on  the  given  line  then  the  abscissas 
Xi  of  PI  and  x^  of  P2  are  known.  Let  us  denote  the  abscissa  of 
P  by  x.  Then  we  have 

PiP  =  x  -  xi,         PiP2  =  x2  -  Xi  ; 


HI,  §47] 


GRAPHIC   REPRESENTATION 


55 


hence  the  abscissa  x  of  P  must  satisfy  the  condition 


(3) 


X  — 


X2  - 


whence  solving  for  x, 

(4)  x  =  Xi  +  \(x2  -  Xi). 

If  the  segments  PiP  and  PiP2  have  the  same  sense,  the 
division  ratio  is  positive  and  P  and  P2  lie  on  the  same  side  of  P\. 
If  the  segments  PiP  and  PiP2  are  oppositely  directed,  then  the 
division  ratio  is  negative  and  P  and  P2  are  on  opposite  sides 
of  PI.  Thus,  if  the  abscissas  of  PI  and  P*  are  2  and  14,  the 
abscissas  of  the  points  that  divide  PiP2  in  the  ratios  3,  ^,  f  , 
-  1,  -  1,  -  2  are  6,  8,  10,  -  4,  -  10,  -  22. 

47.  Point  of  Division.  To  find  the  coordinates  of  the  point 
which  divides  the  line  joining  two  given  points  in  a  given  ratio  \. 

Let  P\(XI,  yi)  and  P2(o;2,  y2)  be  the  two  given  points,  X  the 
given  ratio,  and  P(x,  y)  the  required  point. 

Draw  PiQi,  PQ,  P2Q2  parallel  to  the  y-axis,  and  PiRi,  PR, 
P2R2  parallel  to  the  z-axis.  Then  Q  and  R  divide  Q\Q2  and 
R\Ri,  respectively,  in  the  ratio  X.  Now  as  OQ\  =  Xi,  OQ2  =  x2, 
OQ  =  x,  it  follows  from  (4)  §  46  that 

(5)  x  =  Xi  +  \(x2  -  Xi). 
In  the  same  way  we  find 

(6)  y  =  yi  +  K(y2  -  yi). 

Thus,  the  coordinates  x,  y  of  P  are  expressed  in  terms  of  the 
coordinates  of  PI  and  P2  and  the  division  ratio  X. 


56  MATHEMATICS  [III,  §  48 

48.  Middle  Point.  If  P  be  the  middle  point  of  PiP2,  X  =  |, 
and 

*  =  |0i  +  a*),         y  =  K*/i  +  2/s). 

That  is,  the  abscissa  of  the  mid-point  of  a  segment  is  one  half 
the  sum  of  the  abscissas  of  its  end  points,  and  the  ordinate  is  one 
half  the  sum  of  the  ordinates. 

EXERCISES 

1.  Find  the  lengths  of  the  sides  of  the  following  triangles : 

(a)   (4,  8),  (-  4,  -  8),  (1,  4).  (6)    (4,  5),  (4,  -  5),  (-  4,  5). 

(c)    (2,  1),  (-  1,  2),  (-  3,  0).  (d)   (-  2,  1),  (-  3,  -  4),  (2,  0). 

(e)    (2,  3),  (1,  -  2),  (3,  8).  (/)    (5,  2),  (-  3,  —  2),  (7,  3). 

What  inference  can  be  drawn  from  the  answers  to  (e)  and  (/)  ? 

2.  Find  the  lengths  of  the  sides  and  of  the  diagonals  of  the  quadri- 
lateral (2,  1),  (5,  4),  (4,  7),  (1,  4). 

3.  A  (0,2),  B  (3,  0),  and  C  (4,  8)  are  the  vertices  of  a  triangle. 
Show  that  the  distance  from  A  to  the  mid-point  of  BC  is  one-half  the 
length  of  BC. 

4.  Show  that  two  medians  of  the  triangle  (1,  2),  (5,  5),  (—  2,  6)  are 
equal.     What  inference  can  you  draw? 

5.  The  ends  of  one  diagonal  of  a  parallelogram  are  (4,  —  2)  and 
(—4,  —  4).     One  end  of  the  other  diagonal  is  (1,  2).     Find  the  other 
end. 

6.  The  end  points  of  a  segment  PQ  are  (1,—  3)  and  (5,  0).     Find 
the  length  of  the  segment,  and  the  lengths  of  its  projections  on  the 
x  and  y  axes. 

7.  Show  that  (0,  10),  (1,  1),  (5,  6)  are  the  vertices  of  an  isosceles 
right  triangle. 

8.  Find  the  coordinates  of  the  point 

(a)  Two-thirds  of  the  way  from  (—  1,  7)  to  (8,  1) ; 
(6)  Two-thirds  of  the  way  from  (8,  1)  to  (-  1,  7)  ; 

(c)  Four-sevenths  of  the  way  from  (1,  —  7)  to  (8,  0) ; 

(d)  Three-sevenths  of  the  way  from  (8,  0)  to  (1,  —  7). 

9.  The  segment  from  (4,  5)  to  (2,  3)  is  produced  half  its  length. 
Find  the  end  point. 


Ill,  §50]  GRAPHIC  REPRESENTATION  57 

49.  Locus  of  a  Point  in  a  Fixed  Plane.     If  a  point  is  forced 
to  move  so  as  to  be  always  equidistant  from  two  fixed  points, 
we  know  that  it  must  lie  on  the  perpendicular  bisector  of  the 
segment  joining  these  points.     If  a  point  must  be  at  a  constant 
distance  from  a  fixed  point,  it  will  lie  on  a  circle.     If  a  point 
must  be  always  equidistant  from  a  fixed  point  and  a  fixed  line, 
it  will  lie  on  a  certain  curve,  called  a  parabola,  which  we  have  not 
yet  studied. 

If  x  and  y  are  the  coordinates  of  a  point  P,  the  values  of  x 
and  y  change  as  P  moves  in  the  plane.  For  this  reason  they  are 
called  variables.  If  P  is  subject  to  a  condition  which  forces  it  to 
lie  on  a  certain  curve,  then  x  and  y  must  satisfy  a  certain  condi- 
tion which  can  be  expressed  as  an  equation  in  x  and  y. 

For  example,  if  P  is  always  equidistant  from  (1,  2)  and  (2,  1), 
then,  for  all  positions  of  P,  x  —  y  =  0.  If  P  is  always  equi- 
distant from  (0,  2)  and  the  x-axis  then  x2  —  4y  +  4  =  0.  If  P 
is  always  3  units  from  the  origin,  then  x2  +  yz  =  9. 

Whenever  a  plane  curve  and  an  equation  in  x  and  y  are  so 
related  that  every  point  on  the  curve  has  coordinates  which 
satisfy  the  equation,  and  conversely,  every  real  solution  of  the 
equation  furnishes  coordinates  of  a  point  on  the  curve,  then  the 
equation  is  called  the  equation  of  the  curve,  and  the  curve  is 
called  the  locus  of  the  equation.  This  dual  relation  between 
equation  and  curve  is  the  subject  of  study  in  Analytic  Geometry. 

50.  Equation  of  a  Locus.     To  find  the  equation  of  the  locus 
of  a  point  which  moves  in  a  plane  according  to  some  stated  law, 
we  proceed  as  follows:    First,  draw  a  pair  of  coordinate  axes; 
and  locate  and  denote  by  appropriate  numbers  or  letters  all 
fixed  distances,  including  the  coordinates  of  fixed  points.   Second, 
mark  a  point  P  with  coordinates  x  and  y,  to  represent  the  mov- 
ing point ;  express  the  conditions  of  the  problem  in  terms  of  x,  y, 
and  the  given  constants ;    and  simplify  the  resulting  equation. 


58  MATHEMATICS  [III,  §  50 

Third,  show  that  every  real  solution  of  the  equation  so  obtained 
gives  a  point  which  satisfies  the  conditions  governing  the  motion 
of  P. 

EXAMPLE.  Find  the  equation  of  the  locus  of  a  point  which  is  always 
equidistant  from  a  fixed  line  and  a  fixed  point. 

First.  We  are  free  to  choose  the  axes  where  we  please.  It  is  conven- 
ient to  take  the  fixed  line  for  the  z-axis,  and  to  take  the  y-axis  through 
the  fixed  point.  Then  the  coordinates  of  the  fixed  point  may  be  called 
(0,  a). 

Second.  The  distance  from  P(x,  y)  to  the  fixed  line  is  y,  and  its  dis- 
tance to  the  fixed  point  (0,  a)  is  ^x2  +  (y  —  a)2.  Hence  the  condition 
expressed  in  the  problem  gives  y  =  Vz2  -|-  (y  —  a)2.  This  simplifies  to 

x2  +  2ay  =  a2. 

Third.  It  is  easy  to  show,  by  reversing  the  above  prqcess,  that  if 
x  =  h,  y  =  k,  is  any  solution  of  this  equation,  then  the  point  Q  (h,  k)  is 
equidistant  from  the  re-axis  and  the  point  (0,  a). 

Therefore  x2  +  2ay  =  a2  is  the  required  equation. 

EXERCISES 

1.  Find  the  equation  of  the  locus  of  a  point  which  moves  so  that : 
(a)  itx  is  equidistant  from  the  coordinate  axes ; 

(6)  it  is  four  times  as  far  from  the  z-axis  as  from  the  y-axis ; 

(c)  the  sum  of  its  distances  from  the  axes  is  6 ; 

(d)  the  square  of  its  distance  from  the  rr-axis  is  four  times  its  distance 
from  the  y-axis. 

2.  Find  the  equation  of  the  locus  of  a  point  that  is  always  equi- 
distant from  (4,  -  2)  and  (7,  3).  Ans.   3x  +  5y  =  19. 

3.  Find  the  equation  of  the  perpendicular  bisector  of  the  segment 
joining  the  two  points  (a,  6)  and  (c,  d). 

Ans.    (a  -  c)x  +  (b  -  d)y  =  \(a?  +  b2  -  c2  -  d2}. 

4.  Find  the  equation  of  the  locus  of  a  point  whose  distance  from  the 
point  (—  3,  4)  is  always  equal  to  5.      .4ns.   x2  +  y2  +  6x  —  8y  =  0. 

5.  Find  the  equation  of  the  circle  whose  center  is  (a,  b)  and  whose 
radius  is  c. 


Ill,  §  51] 


GRAPHIC  REPRESENTATION 


59 


51.  Locus  of  an  Equation.  In  general  a  single  equation  in 
x  and  y  has  an  infinite  number  of  real  solutions.  Each  of  these 
solutions  furnishes  the  coordinates  of  a  point  on  the  locus. 

To  find  solutions  and  plot  points  on  the  curve,  solve  the  equa- 
tion, if  possible,  for  y  in  terms  of  x,  or  vice  versa.  Determine  and 
tabulate  a  convenient  number  of  solutions  by  assigning  values 
to  x  and  computing  the  corresponding  values  of  y.  Using  these 
for  coordinates,  plot  the  points  which  they  represent  and  draw 
a  smooth  curve  through  the  plotted  points. 

EXAMPLE  1.     Construct  the  locus  of  the  equation 

x2  =  4(x  +  y). 
Solving  the  given  equation  for  y  we  have 


Assigning  to  x  the  values  0,  1,  2,  3,  etc.,  —  1,  —  2,  —  3,  etc.,  and  com- 
puting the  corresponding  values  of  y,  we  have  the  following  solutions. 


X.  .  . 

0 

1 

2 

3 

4 

5 

6  . 

7 

-  1 

-  2 

-  3 

y.  .  • 

0 

-  .75 

-  1 

-  .75 

0 

1.25 

3 

5.25 

1.25 

3 

5.25 

We  choose  the  axes,  as  in  Fig.  16,  so  that  all  these  points  will  go  on 
the  sheet. 


FIG.  16 


On  plotting  the  points  and  drawing  a  smooth  curve  through  them, 
we  have  a  sketch  of  the  locus  as  shown. 


60 


MATHEMATICS 


[III,  §  52 


EXAMPLE  2.     Plot  the  curve  whose  equation  is 

x2  +  y2  =  Qx  +  2y. 
Solving  the  given  equation  for  y,  we  have 


y  =  1  ±  Vl  +  6z  -  x2, 
and  we  tabulate  solutions  as  follows. 


X.  ... 

0 

1 

2 

3 

4 

5 

6 

7 

-  1 

-  2 

y.  .  .  . 

0 

-  1.45 

2 

-  2.16 

-  2 

-  1.45 

0 

imag. 

imag. 

imag. 

2 

3.45 

4 

4.16 

4 

3.45 

2 

7 


FIG.  16a 


We  note  that  each  value  of  x  gives  two  values  of  y,  i.e.  there  are  two 

points  on  the  curve  having  the  same 
abscissa.  We  find  also  that  values  of 
x  ^  7  do  not  give  real  values  of  y  and 
that  the  same  is  true  for  values  of 
x  ^  —  1. 

When  these  points  have  been  plotted 
and  a  curve  drawn  through  them  we 
have  the  locus  as  shown  in  Fig.  16a. 

52.  Study  of  the  Equation. 
Important  facts  about  the  shape 
and  extent  of  the  locus  can  be 
learned  by  a  study  of  its  equation.  In  the  first  example  above, 
the  equation  is  of  the  first  degree  in  y.  From  this  we  infer  that 
every  value  of  x,  without  exception,  gives  exactly  one  value  of  y. 
Therefore  every  vertical  line  cuts  the  curve  in  one  and  only 
one  point.  As  x  increases  beyond  2,  y  always  increases,  and 
the  curve  goes  off  beyond  all  limit  in  the  first  quadrant.  The 
same  is  true  in  the  second  quadrant.  On  the  other  hand,  the 
equation  is  of  the  second  degree  in  x.  When  solved,  it  gives 

X  =  2  ±  2\/l  +y; 

hence  every  value  of  y  greater  than  —  1  gives  two  real  values 
of  x  but  every  value  of  y  less  than  —  1  gives  an  imaginary  value 
of  x.  Hence  every  horizontal  line  above  y  =  —  1  cuts  the  curve 
in  two  points,  but  there  are  no  points  on  the  curve  below  y  =  —  1. 


Ill,  §  53]  GRAPHIC  REPRESENTATION  61 

The  equation  of  the  second  example,  when  solved  for  y  as 
above,  shows  that  values  of  x  which  make  1  +  6x  —  x2  <  0 
give  imaginary  values  for  y.  Hence  there  are  no  points  on  the 
curve  to  the  left  of  the  line  x  =  3  —  V 10  =  —  0.16,  nor  to  the 
right  of  the  line  x  =  3  +  V 10  =  6.16,  but  every  vertical  line 
between  these  limits  cuts  the  curve  in  two  points. 

If  we  solve  the  same  equation  for  x,  we  find 


x  =  3  ±      Q  +2y  -y2-, 

hence  there  are  no  points  below_the  line  y  =  1  —  VlO  =  —  2.16 
nor  above  the  line  y  =  1  +  VlO  =  4.16,  but  every  horizontal 
line  between  these  lines  cuts  the  curve  in  two  points. 

If  the  equation  is  a  polynomial  in  x  and  y  equated  to  zero, 
a  glance  will  show  whether  or  not  it  passes  through  the  origin. 

The  intercepts  *  can  be  found  by  the  rule :  To  find  the  x-inter- 
cepts  let  y  =0  and  solve  for  x.  Similarly  find  the  7/-intercepts. 

53.  Symmetry.  Two  points  A  and  B  are  said  to  be  sym- 
metric with  respect  to  a  point  P  when  the  line  AB  is  bisected  by  P. 

Two  points  A  and  B  are  said  to  be  symmetric  with  respect  to 
an  axis  when  the  line  AB  is  bisected  at  right  angles  by  the  axis. 

If  the  points  of  a  curve  can  be  arranged  in  pairs  which  are 
symmetric  with  respect  to  an  axis  or  a  point,  then  the  curve 
itself  is  said  to  be  symmetric  with  respect  to  thai  axis  or  point. 

RULE  I.  //  the  equation  of  a  locus  remains  unchanged  in  form 
when  in  it  y  is  replaced  by  —  y,  then  the  locus  is  symmetric  with 
respect  to  the  axis  of  x. 

For,  if  (x,  y}  can  be  replaced  by  (x,  —  y}  throughout  the 
equation  without  affecting  the  locus,  then  if  (a,  6)  is  on  the 

*  The  intercepts  of  a  curve  on  the  axis  of  x  are  the  abscissas  of  the  points  of  inter- 
section of  the  curve  and  the  z-axis.  The  intercepts  on  the  j/-axis  are  the  ordinates  of 
the  points  of  intersection  of  the  curve  and  the  j/-axis. 


62  MATHEMATICS  [III,  §53 

locus,  (a,  —  6)  is  also  on  the  locus,  and  the  points  of  the  locus 
occur  in  pairs  symmetric  with  respect  to  the  axis  of  x. 

We  can  also  prove  the  following  rules. 

RULE  II.  //  the  equation  of  a  locus  remains  unchanged  in 
form  when  in  it  x  is  replaced  by  —  x,  then  the  locus  is  symmetric 
with  respect  to  the  y-axis. 

RULE  III.  //  the  equation  of  a  locus  remains  unchanged  in 
form  when  in  it  x  and  y  are  replaced  by  —  x  and  —  y,  then  the 
locus  is  symmetric  with  respect  to  the  origin. 

54.  Points  of  Intersection.  If  two  curves  whose  equations 
are  given  intersect,  the  coordinates  of  each  point  of  intersection 
must  satisfy  both  equations  when  substituted  in  them  for 
x  and  y.  In  algebra  it  is  shown  that  all  values  satisfying  two 
equations  in  two  unknowns  may  be  found  by  regarding  these 
equations  as  simultaneous  in  the  unknowns  and  solving.  Hence 
the  rule  to  find  the  points  of  intersection  of  two  curves  whose 
equations  are  given. 

Consider  the  -equations  as  simultaneous  in  the  coordinates,  and 
solve  for  x  and  y. 

Arrange  the  real  solutions  in  corresponding  pairs.  These  will 
be  the  coordinates  of  all  of  the  points  of  intersection. 

EXERCISES 
Plot  the  loci  of  the  f ollowing  equations : 

1.  2z  -  3y  -  6  =  0.  12.  4z2  -  ?/2  =  0. 

2.  4z  -  Qy  -  6  =  0.  13.  6z2  +  5xy  -  6y*  =  0. 

3.  6z  -  9y  +  36  =  0.  14.  x2  +  yz  =  4. 

4.  2x  +  3y  +  5  =  0.  15.  x2  -  y2  =  4. 

5.  3x  -  2y  -  12  =  0.  16.  x2  +  y2  =  25. 

6.  5z  +  2y  -  4  =  0.  17.  (x  -  8)2  +  (y  -  4)2  =  25. 

7.  y  =  7x  -  3.  18.  (x  -  4)2  +  (y  -  2)2  =  5. 

8.  2y  -  x  =  2.  19.  4(x  +  !)  =  (?/-  2)2. 

9.  2x  +  9y  +  13  =  0.  20.  10y  =  (x  +  I)2. 

10.  (x  -  4)(y  +  3)  =  0.  21.  y  =  x3  -  4z2  -  4x  +  16. 

11.  (x2  -  4)(y  -  2)  =  0. 


Ill,  §55]  GRAPHIC   REPRESENTATION  63 

22.  y  =  x,  xz,  x3,  x4,  •••,  xn.     What  points  are  common  to  these  curves? 

23.  if  =  x,  x2,  x3,  r».  24.    y  =  (x  -  1),  (x  -  I)2,  (x  -  I)3. 
25.  y  =  (x  -  l)(x  -  2)(x  -  3).      26.    y  =  (x  -  l)(x  -  2)2. 

27.  y  =  (x  -  2)'.  28.    rf  =  (x  -  l)(x  -  2)(x  -  3). 

29.  y"-  =  (x  -  l)(x  -  2)2.  30.    y2  =  (x  -  2)3. 

31.  y  -— £-.  32.    y  = 


x  -  1  x  +  1 

33-    !f----  34.    y- 


x2  +  1  x2  +  1 

^  ~  3> 


35.    y  =        ~  -       .  36.  y  = 

(x  -  2)(x  -  4)  (x  -  2)(x  +  4) 

Find  the  points  of  intersection  of  the  following  curves  : 

I2x  +  y  =  5,  [x-y  =  2, 

'    \x  +  2y  =  l.  38'  t  2*  -3y  =  l. 

+  y'  =  18,  f  x2  +  y2  =  18, 

40'        -3*. 


Ans.    (3,  3),  (-  f,  -  V).  Ans.    (3,  3),  (3,  -  3). 

f  3x2  +  4y2  =  48,  f  3x2  -  4?/2  =  11, 

'  ' 


x  -  y  +  1  =  0.  '    1  4x  =  3y2. 

Ans.    (2,  3),  (-  V,  -  -V)-  ^«s.    (3,  2),  (3,  -  2). 

43.    IX1J  =  2>  44. 

1  y2  =  4x.  x2  +  r/2  -  5. 

45     f  xy  =  x  +  y  +  1,  46     f  xy  =  x  +  ?/  +  1, 


(xy  = 
\4x  - 


?/  =  x  -  1.  I  4x  -  3j/  +  1  =  0. 

Ans.    (3,  2).  Ans.    (2,  3),  (-  J,  -  J). 

47.  Find  the  length  of  the  common  chord  of  the  two  circles  x2  -f-  yz 
=  4x  and  x2  +  y2  =  4(x  +  y-  1).  Ans.   2^3. 

48.  In  what  respects  are  the  loci  of  the  following  equations  sym- 
metric? 

(o)  y  =  x2.  (e)  ?/2  =  x2.  (i)  x3  —  y3  —  x  —  y  =  0. 

(6)  y2  =x.  (/)r/2  =x*.  (T)  XT/  =  a. 

(c)  y  =  x3.  (g)  y  =  Xs  -  x.  (fc)  ax2  +  by2  =  1. 

(d)  y2  =  x3.  (A)  y  =  x4  -  x2.  (I)  ax2  +  26xr/  +  ct/2  =  1. 

55.  Straight  Line  Parallel  to  an  Axis.  Suppose  a  point 
moves  about  on  a  piece  of  coordinate  paper  in  such  a  way  that 
it  is  always  two  units  to  the  right  of  the  axis  of  y.  It  would 


64  MATHEMATICS  [III,  §55 

evidently  be  on  the  line  A  B  that  is  parallel  to  the  y-axis  and 
at  a  distance  of  two  units  to  the  right  of 
OF.  Every  point  of  the  line  AB  has  an 
abscissa  of  two  (x  =  2),  and  every  point 
whose  abscissa  is  two  lies  on  the  line  AB. 
For  this  reason  we  say  that  the  equation 

x  =  2 


I       2  A 

FIG.  17 


represents  the  line  AB  or  is  the  equation  of  the  line  AB. 
More  generally,  the  equation 

x  =  a, 

where  a  is  any  real  number,  represents  a  straight  line  parallel 
to  the  y-axis  and  at  a  distance  a  from  it.  Similarly,  the  equation 
y  =  b  represents  a  line  parallel  to  the  z-axis. 

56.  Straight  Line  through  the  Origin.     Suppose  a  point 
moves  about  on  a  piece  of  coordinate  paper  in  such  a  way  that 
its  distance  from  the  x-axis,  represented  by  y,  is  always  equal 
to  m  times  its  distance  from  the  ?/-axis,  represented  by  x.     The 
equation  of  the  locus  of  the  point  is 

y  =  mx. 

This  is  the  equation  of  a  straight  line  through  the  origin.  The 
points  of  this  line  have  the  property  that  the  ratio  y/x  of  their 
coordinates  is  the  same  number  m,  wherever  on  this  line  the 
point  is  taken.  Moreover  for  any  point  Q,  not  on  this  line, 
the  ratio  y/x  must  evidently  be  different  from  m.  The  number 
m  is  catted  the  slope  of  the  line. 

57.  Proportional  Quantities.     Whenever  two  quantities  y 
and  x  vary  in  such  a  manner  that  their  ratio  y/x  is  always 
constant,  say  m,  they  are  said  to  be  proportional.     The  constant 
m  is  called  the  factor  of  proportionality.     Many  instances  occur 


HI,  §57] 


GRAPHIC   REPRESENTATION 


65 


in  the  applied  sciences  of  two  quantities  related  in  this  manner. 
It  is  often  said  that  one  quantity  varies  as  the  other.     Thus 
Hooke's  law  states  that  the   elon- 
gation   E    of     a     stretched    wire 
or  spring  varies  as  the   tension  t; 
that  is,  E  =  kt,  where  A;  is  a  con- 
stant.    For  a  given  wire,  when  E 
was   expressed   in  thousandths   of 
an  inch  and  t  in  pounds,  the  fol- 
lowing relation  was  found: 

E  =  .8« 
Thus  when  t  =  10,  E  =  8  and  when  t  =  5,  E  =  4. 


5 

FIG.  18 


10      T 


EXERCISES 


Draw  the  lines 


1.  x  =  1,  -  1,  0,  2,  3,  -  2,  -  3,  -  4,  4.       ^ 

2.  y  =  1,  -  1,  0,  2,  3,  -  2,  -  3,  -  4,  4. 

3.  What  is  the  locus  of  a  point  if  x  >  3?     x  =  3?     x  <  3? 

4.  What  is  the  locus  of  a  point  if2<x<3?     2  <  x  <  3?    2<x 
<  3?     2  <  x  <  3? 

5.  What  is  the  locus  of  a  point  if  2  <  x  <  3  and  1  <  y  <  2? 

6.  What  is  the  locus  of  a  point  if  x2  +  y*  <  16  and  x  >  2? 

7.  What  is  the  locus  of  a  point  if  9  <  x2  +  y2  <  16? 

8.  A  stand-pipe  is  filled  at  the  rate  of  150  gallons  per  hour.     What  is 
the  amount  A  of  water  in  the  stand-pipe  h  hours  after  filling  begins? 

9.  A  man  saves  $50  each  month  and  deposits  it  in  a  bank.     What 
is  the  amount  A  which  he  has  in  the  bank  after  t  months? 

10.  A  railroad  track  has  a  rise  of  1  ft.  in  20.     Give  its  equation 
and  plot. 

11.  The  extension  E  in  feet  of  a  spiral  spring  due  to  a  tension  I  of 
1  lb.,  was  1  inch.     What  is  the  relation  connecting  E  and  <?     (Use 
Hooke's  law.) 


6 


66 


MATHEMATICS 


[III,  §58 


58.  Slope  of  a  Straight  Line.  The  slope,  m,  of  the  line 
passing  through  two  points  PI(XI,  yi),  P2(z2,  2/2),  Fig.  19,  is 
given  by  the  formula 


(8) 


m  = 


PiR 


59.  Equation  of  a  Line  through 
two  Points.  Let  the  two  given 
points  be  PI(XI,  yi),  Pz(x2,  y2). 
Let  P(x,  y)  be  any  other  point  on 
the  line  joining  PiP2.  Draw  PiRS 
parallel  to  the  z-axis.  Draw  P\M\, 
P2M2,  PM,  parallel  to  the  y-axis. 


FIG.  19 
Then  since  the  triangles  PiSP  and  PiRP2  are  similar,  we  have 


SP 
PiS 


RPj 
PiR' 


y  - 


?/2  - 


X    —  Xi         X- 

which  may  be  written  in  the  form 
(9)  V*  ~ 


The  equation  of  a  straight  line  with  a  given  slope  m  and 
passing  through  a  given  point   (x\,  y\)  is  seen  from  the  last 
equation  to  be 
(10)  y  -  yi  =  m(x  -  jci). 

In  particular  if  the  y-intercept  is  given  as  b,  the  equation  of 
the  straight  line  having  the  given  intercept  and  with  slope  m  is 


which  reduces  to 
(11) 


y  —  b  =  m(x  —  0) 
y  =  mx  +  b. 


HI,  §61] 


GRAPHIC  REPRESENTATION 


67 


This  last  equation  is  called  the  slope  form  of  the  equation  of 
the  straight  line. 

If  both  intercepts  are  given,  say  Z -intercept  =  a,  ^/-intercept 
=  b,  we  can  find  the  equation  of  the  line  by  means  of  the 
equation  for  a  line  through  two  given  points.  We  have 


which  reduces  to 
(12) 


This  is  called  the  intercept  form  of  the  equation  of  the  straight 
line. 

60.  Parallel  Lines.  Con- 
sider two  parallel  lines  PiRi 
and  PzRz-  Draw  R2Ri  and  PzPi 
parallel  to  the  ?/-axis,  and  RiSi, 
RzSz  parallel  to  the  x-axis. 
Then  since  the  triangles  RiSiPi 
and  RiSiPz  are  equal, 


and 


i  =  SZPZ. 


FIG.  20 


Hence, 


That  is  the  slopes  of  any  two  non-vertical  parallel  lines  are  equal. 
61.  Perpendicular  Lines.  Consider  two  perpendicular  lines 
LI  and  Z/2  intersecting  at  P\(x\,  t/i).  Let  PI(XI  +  a,  y\  +  &)  be 
a  second  point  on  L\\  then  since  the  given  lines  are  perpen- 
dicular, the  point  Qi(x\  —  b,  yi  +  a)  lies  on  L2  as  shown  by 
construction  in  the  figure.  Then  the  slope  of  L\  is  mi  =  b/a, 
by  the  definition  of  slope,  §  58;  and  the  slope  of  L2  is  m2  =* 


68 


MATHEMATICS 


[HI,  §61 


—  (a/6),  for  the  same  reason.     It  follows  that  we  have 
(13)  mim-j  =  —  1. 


This  proves  the  theorem: 
//  two  non-vertical  lines  are 
perpendicular,  then  the  prod- 
uct of  their  slopes  is  —  1. 

The  converse  is  also  true: 
//  the  product  of  the  slopes 
of  two  lines  is  —  1,  then 
they  are  perpendicular.  The 
proof,  which  is  suggested 
by  Fig.  21,  is  left  to  the 
student. 

62.  General  Equation  of  the  First  Degree.     The  equation 


V 

FIG.  21 


(14) 


Ax  +  By  +  C  =  0, 


where  A,  B,  C  are  constants,  is  called  the  general  equation  of 
the  first  degree  in  x  and  y  because  every  equation  of  the  first 
degree  may  be  reduced  to  that  form.  For  any  values  what- 
soever of  A,  B,  and  C,  provided  A  and  B  are  not  both  zero, 
the  general  equation  of  the  first  degree  represents  a  straight 
line. 

EXERCISES 

1.  Find  the  slope  of  the  line  joining  the  points 

(a)   (1,  3)  and  (2,  7).  (6)   (2,  7)  and  (-  4,  -  4). 

(c)    ( A/3,  V2)  and  (-  >/2,  >/3).          (d)   (a  +  b,  c  +  a)  and  (c  +  a,  6+c). 

2.  Prove  by  means  of  slopes  that  (-  4,  -  2),  (2,  0),  (8,  6),  (2,  4) 
are  the  vertices  of  a  parallelogram. 

3.  Prove  by  means  of  slopes  that  (0,  -  2),  (4,  2),  (0,  6),  (-  4,  2) 
are  the  vertices  of  a  rectangle. 

4.  What  are  the  equations  of  the  sides  of  the  figures  in  Exs.  2 
and  3. 


Ill,  §62]  GRAPHIC   REPRESENTATION  69 

5.  Find  the  intercepts  and  the  slope  of  each  of  the  following 
lines: 

(a)  2x  +  3y  =  6.  (6)  x  -  2y  +  5  =  0. 

(c)  3z  -  y  +  3  =  0.  (d)  5x  +  2y  -  6  =  0. 

(e)   7x  -  4y  -  28  =  0.  (/)  3y  -  2x  =  8. 

6.  Find  the  equations  of  the  lines  satisfying  the  following  conditions: 
(a)  passing  through  (—3,  1)  and  slope  =  2. 

(6)  having  the  ^-intercept  =  3,  y-intercept  =  —  2. 

(c)  slope  =  —  3,  x  intercept  =  4. 

(d)  x  intercept  =  —  3,  y  intercept  =  —  4. 

(e)  passing  through  the  point  (2,  3)  and  with  slope  =  —  2. 

7.  Find  the  points  of  intersection  of 

(a)  x  -  7y  +  25  =  0,     z2  +  y2  =  25. 

(b)  2x2  +  3y2  =  35,     3z2  -  4y  =  0. 

(c)  x2  +  y  =  7,     y*  -  x  =  7. 

(d)  y  =  x  +  5,     9z2  +  16y2  =  144. 

8.  Find  the  equations,  and  reduce  them  to  the  general  form,  of  the 
lines  for  which 

(a)  m  =  2,         b  =  -  3.  (6)  m  =  -  1/2,     b  =  3/2. 

(c)  m  =  2/5,      b  =  -  5/2.  (d)  m  =  1,  b  =  -  2. 

(e)  a  =  3,          6  =  3.  (/)  a  =  4,  6  =  2. 
(0)  a  =  -  3,     6  =  -  3.  (h)    a  =  4,  &  =  -  2. 
(t)  a  =  -  3,     b  =  3.  0')     a  =  2,             6=4. 

9.  Write  the  equations  of  the  lines  passing  through  the  points: 
(a)  (-  2,  3),   (-  3,  -  1).  (b)   (5,  2),  (-  2,  4). 

(c)    (1,  4),  (0,  0).  (rf)  (2,  0),  (-  3,  0). 

(e)    (0,  2),   (3,  -  1).  (/)  (2,  3),  (-  6,  -  5). 

10.  Write  the  equations  of  the  lines  passing  through  the  given 
points  and  with  the  given  slopes: 

(a)   (-2,3),     m  =  2.  (6)   (5,2),          m  =  1. 

(c)    (1,  4),         m  =  i  (d)  (2,  0),         m  =  -  f. 

(e)   (0,  2),         w  =  0.  (/)  (3,  -  2),     m  =  -  2. 

11.  Write  the  equation  of  the  line  which  shall  pass  through  the 
intersection  of  2y  +  2x  +  2  =  0  and  3y  —  x  —  8  =  0,  and  having  a 
slope  =  4.  Am.  IQx  —  4y  +  51  =0. 


70  MATHEMATICS  [III,  §62 

12.  What  are  the  equations  of  the  diagonals  of  the  quadrilateral 
the  equations  of  whose  sides  are  y  —  x  +  1  =  0,  y  =  —  x  +  2,  y  =  3x 
+  2,  and  y  +  2x  +  2  =0? 

13.  Required  the  equation  of  the  line  which  passes  through  (2,  —  1) 
and  is 

(a)  parallel  to  2y  -  3x  -  5  =  0.  Am.  2y  -  3x  +  8  =  0. 

(6)  perpendicular  to  2y  —  3x  —  5  =  0.  Ans.  2x  +  3y  —  1  =0. 

14.  Find  the  equations  of  the  two  straight  lines  passing  through 
the  point  (2,  3),  the  one  parallel,  the  other  perpendicular  to  the  line 
4x  -  3y  =  6.  Ans.  4x  -  3y  +  1  =  0,  3x  +  4y  -  18  =  0. 

15.  Passing   through    (4,  —  2),    the   one   parallel,    the   other   per- 
pendicular to  the  line  y  =  2x  +  4.     Ans.  y  =  2x  —  10,  x  +  2y  =  0. 

16.  Passing  through  the  point  of  intersection  of  4x  +  y  +  5  =  0 
and  2x  —  3y  +  13  =  0,  one  parallel,  the  other  perpendicular  to  the 
line  through  the  two  points  (3,  1)  and  (—  1,  —  2). 

Ans.  3x  -  4y  +  18  =  0,  4z  +  3y  -  1  =0. 

17.  Find  the  equation  of  the  line  joining  the  origin  to  the  point  of 
intersection  of  2z  +  5r/  —  4  =  0  and  3x  —  2y  +  2  =  0. 

Ans.  y  =  —  8x. 

18.  Find  the  equation  of  the  straight  line  passing  through  the 
point  of  intersection  of  2x  +  5y  —  4  =  0  and  2x  —  y  +  1  =0  and 
perpendicular  to  the  line  5x  —  Wy  =  17.  Ans.  6x  +  3y  =  2. 

19.  Find  the  equations  of  the  lines  satisfying  the  following  condi- 
tions : 

(a)  through  (2,  3),  parallel  to  y  =  7x  +  3. 

(6)  through  (4,  —  1),  perpendicular  to  2x  +  3y  =  6. 

(c)  through  (—2,  -  1),  parallel  to  3y  —  2x  =  1. 

(d)  through  (3,  -  6),  parallel  to  2y  +  4x  =  7. 

(e)  through  (—  1,  —  1),  perpendicular  to  x/2  +  y/3  =  1. 
(/)  through  (2,  2),  perpendicular  to  y  =  —  3x  -f  2. 

20.  Prove  that  the  diagonals  of  a  parallelogram  bisect  each  other. 

21.  Prove  that  the  diagonals  of  a  rhombus  bisect  each  other  at  right 
angles. 

22.  Prove  that  the  diagonals  of  a  square  are  equal  and  bisect  each 
other  at  right  angles. 

23.  A  straight  line  makes  an  angle  of  45°  with  the  x-axis  and  its  y 
intercept  =  2;  what  is  its  equation?  Ans.  y  =  x  +  2. 


Ill,   §62] 


GRAPHIC   REPRESENTATION 


71 


24.  The  following  data  gives  the  height  of  a  plant  in  inches  on 
different  days. 


Height  
Day 

0 
0 

28 
40 

33 
60 

36 

80 

40 
100 

52 
120 

62 
140 

66 
160 

Find  the  rate  of  growth  after  60  days. 

Find  the  rate  of  growth  after  110  days. 

[The  rate  of  growth  is  the  slope  of  the  curve.  The  slope  of  a  curve 
at  a  given  point  is  defined  to  be  the  slope  of  the  tangent  line  drawn  to 
the  curve  at  the  given  point.  Draw  the  tangent  with  a  ruler  and 
with  the  aid  of  the  eye.]  Ans.  7/10  in.  per  day;  0.55  in.  per  day. 


CHAPTER  IV 

LOGARITHMS 

63.  Definitions  and  Preliminary  Notions.    In   the    equa- 
tion 

102  =  100, 

three  numbers  are  involved.  By  omitting  each  number  in  turn 
there  arise  three  different  problems.  If  we  omit  the  100,  we 
have  the  familiar  question  in  involution: 

102  =  ?. 
If  we  omit  the  10  we  have  the  familiar  question  in  evolution: 

?2  =  100, 
or,  as  it  is  usually  written, 

VlOO  =  ?. 

If  we  omit  the  2  we  have  the  following  question 

10?  =  100, 

which  we  agree  to  write  in  the  form, 

\ 

logic  100  =  ? 

and  we  say  that  2  =  the  logarithm  of  100  to  the  base  10. 
In  general,  if 

(1)  IP  =  N, 

then  x  =  the  logarithm  of  N  to  the  base  b,  and  we  write, 

(2)  x  =  logb  N. 

(1)  and  (2)  are  then  simply  two  different  ways  of  expressing 
the  same  relation  between  b,  x,  and  N.  (1)  is  called  the  ex- 

72 


IV,  §63]  LOGARITHMS  73 

ponential  form.     (2)  is  called  the  logarithmic  form.     Either  of 
the  statements   (1)   or  (2),  implies  the  other.     The  exponent 
in  (1)  is  the  logarithm  in  (2),  a  fact  which  may  be  emphasized 
by  writing 
(3)  (base)10*arithm  =  number. 

For  example,  the  following  relations  in  exponential  form: 
32  =  9,        24  =  16,         (1/2)3  =  1/8,        a"  =  x, 
are  written  respectively  in  the  logarithmic  form: 

2  =  logs  9,        4  =  Iog2  16,        3  =  logi/z  1/8,        y  =  logo  x. 

We  shall  now  give  the  following 

DEFINITION  OF  A  LOGARITHM.  The  power  to  which  a  given 
number  called  the  base  must  be  raised  to  equal  a  second  number  is 
called  the  logarithm  of  the  second  number. 

EXERCISES 

1.  Write  the  following  equations  in  logarithmic  form: 
(a)        9  =  32.  (g)     7  =  71.  _ 

(6)       64  =  43.  (h)  25  =  (Vs)4. 

(c)  16  =  24.  (i)     8  =  (V2)«. 

(d)  243  =  35.  0')     3  =  (V3)2. 
(c)       64  =  2".  (fc)     3  -  V9. 
(/)  2401  =  74.  (I)     4  =  v/64. 

2.  Write  the  following  equations  in  exponential  form : 

(a)  log,  16   =  4.  (g)  logioO.l    -  -  1. 

(b)  Iog4  16  =  2.  (A)  logz   1/4  : 2. 

(c)  logio  1000  =  3.  (t)       Iog64  2  =  1/6. 

(d)  logs  729  =  5.  (j)   Iog2   1/8  =  -  3. 
(c)      logs  625  =  4.  (fc)      logn  1  =  0. 
(/)    log»  1728  =  3.                              (0        loga  o  =  l. 

3.  Find  the  numerical  value  of  each  of  th    following : 
(a)  Iog2   64.  (e)   Iog26  5. 

(6)  logio  0.001.  (/)   3  Iog6  625  +  logj  16. 

(c)  Iog27  3.  (g)  logi/2  4. 

(d)  logio  100  -  |  logo.i  100.  (h)  5  logz  16-2  log«  625. 


74  MATHEMATICS  [IV,  §64 

64.  Properties  of  Logarithms.  Any  positive  number,  ex- 
cept 1,  may  be  the  base  of  a  system  of  logarithms  of  all  the  real 
positive  numbers.  In  any  such  system, 

1)  The  logarithm  of  1  is  zero. 
For,  6°  =  1,  therefore  logb  1=0. 

2)  The  logarithm  of  the  base  itself  is  1. 
For,  61  =  6,  therefore  logb  6  =  1. 

3)  The  logarithm  of  a  product  is  the  sum  of  the  logarithms  of 
the  factors. 

For  if  logb  M  =  k  and  logb  -Y  =  I,  then  M  —  bk  and  N  =  bl, 
MN  =  bk-bl  =  bk+l,  whence 

logb  MN  =  k  +  I  =  logb  M  +  logb  N. 

This  can  readily  be  extended  to  three  or  more  factors. 

4)  The  logarithm  of  a  quotient  is  equal  to  the  logarithm  of  the 
dividend  minus  the  logarithm  of  the  divisor. 

For, 

'  M  =y 
N  ~  bl  '' 
therefore 

logb  jf  =  k  -  I  =  logb  M  -  logb  N. 

5)  The  logarithm  of  the  reciprocal  of  a  number  is  the  negative 
of  the  logarithm  of  the  number. 

For  on  putting  M  —  1  under  (4)  above,  we  have 

logb-r^  =  logb  1  -  logb  N  =  -  logb  N, 

since  logb  1=0. 

6)  The  logarithm  of  the  pth  power  of  a  number  is  found  by 
multiplying  the  logarithm  of  the  number  by  p. 

For,  N  =  bk  and  Np  =  (bk)p  =  bpk,  whence 

logb  Np  =  pk  =  p  logb  N. 


IV;   §64]  LOGARITHMS  75 

7)   The  logarithm  of  the  rth  root  of  a  number  is  found  by  dividing 
the  logarithm  of  the  number  by  r. 

For,  N  =  bk  and   A/A7  =  Nllr  =  (6*)1/r  =  bklr,  whence 


logs  VAT  =  -  = 


N 


r 

EXERCISES 

Express  the  logarithms  of  the  following  numbers  in  terms  of  the 
logarithms  of  integers.  In  this  book,  when  the  base  is  omitted,  10  is 
to  be  understood  as  the  base. 

352/3  17i/4  12-2 

!•  ]°g  iQ2/3.Ai/2  •  2.  log  ,,-7,— .  3.  log 


132/3 .  6i/2  • 


4.  Prove  that  logs  V81V729-9-2'3  =  31/18. 

Express  the  logarithms  of  the  following  in  terms  of  the  logarithms 
of  prime  numbers. 

(63)1/4  88~1/2 

(25)2(72)1/4 '  (75)3/4(12)2  * 

7.  log  ^-     — 3.  8.  log  (V2!V72V6). 

9.  Given  log  2  =  0.3010,  log  3  =  0.4771,  log  7  =  0.8451,  find  the 
logarithms  of  the  following  numbers. 

(a)  6.  (e)   32.  (i)  420.  (m)  Vf/2. 

(6)  14.  (/)10.5.  0')  900.  (n)    A/504. 

(c)  24.  (g)   14?.  (k)  35/48.  (o)    BVl3.5. 

(d)  28.  (h)  2.52.  (I)  1/36.  (p)    >/294. 

10.  Express  the  logarithms  of  each  of  the  following  expressions  to 
the  base  a  in  terms  of  logo  b,  loga  c,  Iog0  d. 

11.  Prove  that 


=  2  loga   (X  +  V-l). 

12.  If  log  3  =  0.4771,  what  is  the  (a)  log  30?  (b)  log  300?  (c)  log 
3000?  (d)  log  30,000?  What  part  of  these  logarithms  is  the  same? 
Why? 


76  MATHEMATICS  [IV,  §65 

65.  Computation  of  Common  Logarithms.  While  any 
positive  number  except  unity  could  be  used  as  the  base  of  a 
system  of  logarithms,  only  two  systems  are  in  general  use. 
One,  called  the  natural,  or  Napierian  system  is  used  in  analytical 
work  and  has  the  number  e  =  2.71828  +  for  its  base.  The 
other,  known  as  the  common,  or  Briggs  system  is  used  for  all 
purposes  involving  merely  numerical  computations  and  has  for 
its  base  the  number  10.  Unless  specifically  stated  to  the 
contrary  the  common  system  will  be  the  one  used  throughout 
this  book. 

In  the  following  discussion  of  common  logarithms,  log  x  is 
written  as  an  abbreviation  of  logio  x. 

Every  positive  number  has  a  common  logarithm,  and  the 
value  of  this  logarithm  may  be  obtained  correct  to  as  many 
places  of  decimals  as  may  be  desired.  Negative  numbers  and 
zero  have  no  real  logarithms. 

If  we  extract  the  square  root  of  10,  the  square  root  of  the 
result  thus  obtained,  and  so  on,  continuing  the  reckoning  in 
each  case  to  the  fifth  decimal  figure,  ,we  obtain  the  following 
table : 


1Q1/2 
l0l/4 
1Q1/8 
1Q1/16 

101/32 
1Q1/64 

=  3.16228, 

=  1.77828, 
=  1.33352, 
=  1.15478, 
=  1.07461, 
=  1.03663, 

1Q1/128      _ 
1Q1/256      _ 
101/512      = 
1Q1/1024    _ 
1Q1/2048    _ 
1Q1/4096    = 

1.01815, 
1.00904, 
1.00451, 
1.00225, 
1.00112, 
1.00056, 

and  so  on.     The  exponents  5,  |,  •  •  •  on  the  left  are  the  logarithms 
of  the  corresponding  numbers  on  the  right. 

By  the  aid  of  this  table  we  may  compute  the  common  logar- 
ithm of  any  number  between  1  and  10,  and  hence  of  any  positive 
number. 


IV,  §66]  LOGARITHMS  77 

EXAMPLE.     Find  the  common  logarithm  of  4.26. 

Divide  4.26  by  the  next  smaller  number  in  the  table,  3.16228.  The 
quotient  is  1.34719.  Hence  4.26  =  3.16228  X  1.34719.  Divide 
1.34719  by  the  next  smaller  number  in  the  table,  1.33352.  The  quo- 
tient is  1  0102.  Hence  4.26  =  3.16228  X  1.33352  X  1.0102.  Con- 
tinue thus,  always  dividing  the  quotient  last  obtained  by  the  next 
smaller  number  in  the  table.  We  shall  obtain  by  this  method  an  ex- 
pression for  4.26  in  the  form  of  a  product: 

4.26  =  3.16228  X  1.33352  X  1.00904  X  •  •  • 
=  101/2  X  101/8  X  101/256  X  •  •  • 

Therefore, 


=  .5000 
+  .1250 
+  .0039 

=  .6289 

By  referring  to  the  table  of  logarithms  at  the  end  of  the  book  we  find 
that,  correct  to  four  decimal  places, 

log  4.26  =  .6294 

Hence,  by  using  only  three  terms  in  the  above  approximation  we  ob- 
tain a  result  which  is  in  error  but  5  units  in  the  fourth  decimal  place. 

66.  Characteristic  and  Mantissa.     If  two  numbers  are  un- 
equal, their  logarithms  are  unequal  in  the  same  sense;  that  is  if 

a  <  b  <  c, 
then 

log  a  <  log  b  <  log  c. 
For  example 

log  100  <  log  426  <  log  1000, 
that  is, 

2  <  log  426  <  3. 


78  MATHEMATICS  [IV,  §66 

When  the  logarithm  of  a  number  is  not  an  integer  it  may 
be  represented  approximately  by  a  decimal  fraction  correct  to 
any  desired  number  of  places;  thus  log  426  =  2.6294  to  four 
decimal  places. 

The  integral  part  of  the  logarithm  is  called  the  characteristic 
and  the  decimal  part  is  called  the  mantissa.  In  log  426,  the 
characteristic  is  2  and  the  mantissa  is  .6294.  For  convenience 
in  computing  it  is  desirable  to  have  the  mantissa  positive  even 
when  the  logarithm  is  a  negative  number.  For  example, 
log  \  =  -  0.3010,  but  -  0.3010  =  9.6990  -  10,  and  we  write 

log  \  =  9.6990  -  10, 

in  which  the  characteristic  is  9  —  10  =  —  1,  but  the  mantissa 
.6990  is  positive. 

It  is  convenient  to  write  the  logarithm  of  any  number  N  in 
the  form 

log  #  =  M  -  A;- 10, 

in  which  M  is  a  positive  number  or  zero  and  A;  is  a  positive 
integer  or  zero. 

For  example,  log  426  =  2.6294,  log  42.6  =  1.6294,  log  4.26 
=  0.6294,  log  0.426  =  9.6294  -  10,  log  0.0426  =  8.6294  -  10, 
log  0.00426  =  7.6294  -  10. 

Moving  the  decimal  point  n  places  to  the  right  (left)  in  a  number 
increases  (decreases)  the  characteristic  of  its  common  logarithm 
by  n,  but  does  not  affect  its  mantissa. 

For  this  has  the  effect  of  multiplying  (dividing)  the  number 
by  10",  and 

log  (N  -10")  =  log  N  +  log  10"  =  log  N  +  n 
and 

log  (N  •*•  10")  =  log  N  -  n. 

Therefore,  the  mantissa  of  the  common  .logarithm  of  a  number 
is  independent  of  the  position  of  the  decimal  point.  In  other 


IV,  §66]  LOGARITHMS  79 

words,  the  common  logarithms  of  two  numbers  which  contain 
the  same  sequence  of  figures  differ  only  in  their  characteristics. 
Hence,  tables  of  logarithms  of  numbers  contain  only  the  man- 
tissas and  the  computer  must  determine  the  characteristics 
mentally.  This  can  be  done  by  the  following  simple  rules. 

RULE  I.  The  characteristic  of  the  common  logarithm  of  any 
number  greater  than  1,  is  one  less  than  the  number  of  digits  before 
the  decimal  point. 

For  if  N  is  a  number  having  n  digits  in  the  integral  part  (i.  e. 
before  the  decimal  point),  then 

10n-i  <  N  <  10n 

and 

n  —  1  ^  log  N  <  n; 

therefore  log  N  =  (n  —  1)  -f  (a  decimal  fraction)  and  its 
characteristic  is  n  —  1. 

On  the  other  hand  if  N  is  a  decimal  fraction  (i.  e.,  a  positive 
number  less  than  1),  we  may  move  the  decimal  point  10  places 
to  the  right  and  apply  Rule  I.,  provided  we  subtract  10  from 
the  resulting  logarithm.  For  example, 

log  0.0006958  =  log  6958000  -  10 

and  by  Rule  I.  the  characteristic  is  6  —  10. 

This  process  is  easily  seen  to  be  equivalent  to  that  specified  in 

RULE  II.     To  find  the  characteristic  of  the  common  logarithm 

of  a  number  less  than  1,  subtract  from  9  the  number  of  ciphers 

between  the  decimal  point  and  the  first  significant  figure.     From 

the  number  so  obtained  subtract  10. 

.  A  very  large  number  such  as  the  distance  in  feet  from  the 
earth  to  the  sun,  490,000,000,000  (correct  to  two  significant 
figures),  is  conveniently  written  (on  moving  the  decimal  point 
11  places  to  the  left)  in  the  form 

4.9  X  1011 


80  MATHEMATICS  [IV,  §66 

and  the  characteristic  of  its  common  logarithm  is  11.  Similarly 
a  very  small  number  such  as  0.000,000,453,8  can  be  written 
(on  moving  the  decimal  point  7  places  to  the  right), 

4.538  X  10-7 

and  the  characteristic  of  its  logarithm  is  —  7  =  3  —  10. 

This  form  of  expression  is  frequently  used  where  only  a  few 
significant  figures  are  known  to  be  correct,  and  if  the  decimal 
point  is  placed  after  the  first  significant  figure,  the  exponent  of 
10  is  the  characteristic  of  the  logarithm  of  the  number. 

EXERCISES 
Find  the  characteristics  of  the  logarithms  of  the  following  numbers: 

(1)  276.35  (5)  0.00072  (9)  73.187 

(2)  0.0495  (6)  4589.5  (10)  8.421  X  lO"26. 

(3)  1.837  (7)  0.9372  (11)  7.268  X  101*. 

(4)  6.3  X  10s.  (8)  7.32  X  10~5.  (12)  0.00008 

67.  Use  of  Tables.  1)  The  characteristic  is  not  given  in  the 
table  of  logarithms.  It  is  to  be  found  by  the  above  two  rules. 
It  should  be  written  down  first,  and  always  expressed  even 
though  it  be  zero,  in  order  to  avoid  error  due  to  forgetting  it. 

2)  The  mantissa  of  the  common  logarithms  of  numbers, 
correct  to  four  decimal  places,  are  printed  in  Table  I.,  at  the 
end  of  the  book.  For  convenience  in  printing  the  decimal  points 
are  omitted. 

To  find  the  mantissa  of  a  number  consisting  of  one,  two,  or 
three  digits  (exclusive  of  ciphers  at  the  beginning  or  end,  and 
the  decimal  point),  look  in  the  column  marked  N  for  the  first 
two  digits  and  select  the  column  headed  by  the  third  digit; 
the  mantissa  will  be  found  at  the  intersection  of  this  row  and 
this  column.  For  example,  to  find  the  mantissa  of  456,  we  run 
down  the  column  headed  N  to  45  and  then  run  across  the  page 


IV,  §68]  LOGARITHMS  81 

to  the  column  headed  6  where  we  find  the  mantissa  .6590;  again, 
the  mantissa  of  720  is  found  opposite  72  in  the  column  headed  0, 
and  is  8573. 

EXERCISES 
Look  up  the  following  logarithms  in  Table  I. 

(1)  log  276  =  2.4409  (11)  log  .00782 

(2)  log  8.64  =  0.9365  (12)  log  .0856 

(3)  log  .829  =  9.9186  -  10.  (13)  log  20. 

(4)  log  7.34  X  105  =  5.8657  (14)  log  8.5 

(5)  log  2.30  X  10-3  =  7.3617  -  10.     (15)  log  1870. 

(6)  log  24700  =  4.3927  (16)  log  3.20  X  10~12. 

(7)  log  3.7  X  1012.  .(17)  log  5.47  X  1023. 

(8)  log  9.  (18)  log  7.58  X  10*. 

(9)  log  846000.  (19)  log  98.3 
(10)  log  .000172  (20)  log  3140000. 

68.  Interpolation.  If  there  are  more  than  three  significant 
figures  in  the  given  number,  its  mantissa  is  not  printed  in  the 
table;  but  it  can  be  found  approximately  by  the  principle  of 
proportional  parts:  when  a  number  is  changed  by  an  amount 
which  is  very  small  in  comparison  with  the  number  itself,  the  change 
in  the  logarithm  of  the  number  is  nearly  proportional  to  the  change 
in  the  number  itself. 

For  example,  to  find  the  logarithm  of  37.68,  we  find  from  the 
table, 

Mantissa  of  3760  =  5752, 
Mantissa  of  3770  =  5763. 

The  difference  between  these  mantissas,  called  the  tabular 
difference,  is  11.  We  note  that  an  increase  of  10  in  3760  pro- 
duces an  increase  of  11  in  its  mantissa  and  we  conclude  that  an 
increase  of  8  in  3760  (to  bring  it  up  to  3768,  the  given  digits) 
would  produce  an  increase  of  .8  X  11  =  8.8  in  the  mantissa. 
This  number  8.8,  called  the  correction,  is  to  be  added  to  the 
7 


82  MATHEMATICS  [IV,  §68 

mantissa  of  3760,  but  in  using  a  four  place  table  we  retain  only 
four  places  in  corrected  mantissas,  so  here  we  add  9  (the  integer 
nearest  to  8.8) ;  thus, 

log  37.60  =  1.5752 
correction  =  9 

log  37.68  =  1.5761 

Near  the  beginning  of  Table  I.  the  tabular  differences  are  so 
large  as  to  make  this  process  of  interpolation  inconvenient  and 
in  some  instances  unreliable.  On  this  account  there  are  printed 
on  the  third  and  fourth  pages  of  Table  I.,  the  mantissas  of  all 
four  figure  numbers  whose  first  digit  is  1.  By  using  these  we 
can  avoid  interpolation  at  the  beginning  of  the  table.  Thus, 
on  the  third  page  of  the  table  we  find, 

log  103.2  =  2.0137, 

but  if  we  find  it  by  interpolation  on  the  first  page, 
log  103.2  =  2.0136 

EXAMPLE  1.  Find  the  logarithm  of  .003467.  Opposite  34  in 
column  6  find  5391;  the  tabular  difference  is  12;  .7  X  12  =  8.4;  the 
mantissa  is  then  5391  +  8  =  5399;  hence  log  .003467  =  7.5399  -  10 

EXAMPLE  2.  Find  log  2.6582.  Opposite  26  in  column  5  find  4232; 
the  tabular  difference  is  17;  .82  X  17  =  13.9;  the  mantissa  is  4232 
+  14  =  4246;  hence  log  2.6582  =  0.4246. 

69.  Accuracy  of  Results.  The  accuracy  of  results  obtained 
by  means  of  logarithms  depends  upon  the  number  of  decimal 
places  given  in  the  tables  that  are  used,  and  this  accuracy  has 
reference  to  the  significant  figures  counted  from  the  left.  In 
general,  a  table  will  give  trustworthy  results  to  as  many  sig- 
nificant figures,  counted  from  the  left,  as  there  are  decimal 
places  given  in  the  logarithms.  For  example,  four-place 
logarithms  would  show  no  difference  between  35492367  and 
35490000. 


IV,  §70]  LOGARITHMS  83 

Neither  a  four-place  nor  a  five-place  table  would  be  of  any 
use  in  financial  computations  where  large  sums  are  involved. 
It  would  take  a  nine-place  table  to  yield  exact  results  if  the 
sums  involved  should  reach  a  million  dollars. 

70.  Reverse  Reading  of  the  Table.  To  find  the  number 
when  its  logarithm  is  known.  This  is  sometimes  called  finding 
the  antilogarithm.  For  this  process  we  have  the  following  rule. 

RULE  III.  //  the  mantissa  is  found  exactly  in  the  table,  the 
first  two  figures  of  the  corresponding  number  are  found  in  the 
column  N  of  the  same  row,  while  the  third  figure  of  the  number  is 
found  at  the  top  of  the  column  in  which  the  mantissa  is  found. 

Place  the  decimal  point  so  that  the  rules  in  §  66  are  fulfilled. 

EXAMPLE.     Given  log  N  =  1.7427;  to  find  N. 

We  find  the  mantissa  7427  in  the  row  which  has  55  in  column  N. 
The  column  in  which  7427  is  found  has  3  at  the  top.  Thus  the  sig- 
nificant figures  in  the  number  are  553.  Since  the  characteristic  is  1  we 
must  have  2  figures  to  the  left  of  the  decimal  point.  Thus  N  =  55.3. 

If  the  mantissa  of  the  given  logarithm  is  between  two  man- 
tissas in  the  table,  we  may  find  the  number  whose  logarithm 
is  given  by  the  following 

RULE  IV.  When  the  given  mantissa  is  not  found  in  the  table, 
write  down  three  digits  of  the  number  corresponding  to  the  mantissa 
in  the  table  next  less  than  the  given  mantissa,  determine  a  fourth 
digit  by  dividing  the  actual  difference  by  the  tabular  difference, 
and  locate  the  decimal  point  so  that  the  rules  for  characteristics  are 
fulfilled. 

EXAMPLE.     Given  log  N  =  0.4675;  to  find  N. 

The  mantissa  4675  is  not  recorded  in  the  table,  but  it  lies  between 
the  two  adjacent  mantissas  4669  and  4683.  The  mantissa  4669  corre- 
sponds to  the  number  293.  The  tabular  difference  is  14.  The  actual 
difference  between  4669  and  4675  is  6.  The  number  4675  is  6/14  of 
the  interval  from  4669  to  4683,  and  the  corresponding  number  N  is 


84  MATHEMATICS  [IV,  §70 

about  6/14  of  the  way  from  293  to  294,  or,  reducing  6/14  to  a  decimal, 
about  .4  of  a  unit  beyond  293.  Hence  the  corresponding  digits  are 
2934;  hence  TV  =  2.934. 

The  work  may  be  written  down  as  follows: 
log  N  =  0.4675 
4669 


14)60(4 
N  =  2.934 

EXERCISES 

Obtain  the  logarithm  of  each  of  the  following  numbers. 

1.  3.1416                        2.  1.732  3.  2.718 

4.  1.414                          5.  39.37  6.  0.4343 

7.  3437                           8.  0.0254  9.  0.9144 

10.  0.003954                   11.  0.016018  12.  0.0283 

13.  7918.                         14.  866500.  15.  92897000. 

Find  the  antilogarithm  of  each  of  the  following  numbers. 


16. 

0.4563 

17. 

96378 

-  10. 

18. 

5.3144 

19. 

1.7581 

20. 

8.2046 

-  10. 

21. 

6.1126 

22. 

0.4971 

23. 

7.5971 

-  10. 

24. 

4.9365 

25. 

4.6856  - 

10. 

26. 

8.1530 

-  10. 

27. 

8.6123 

-  20. 

28. 

8.4048  - 

10. 

29. 

8.4520 

-  10. 

30. 

0.7318 

-  20. 

71.  Cologarithms.     The    cologarithm   of   a   number   is   the 
logarithm  of  the  reciprocal  of  the  number.     (Compare  (5)  §  64.) 

Thus  colog  425  =  log  — -  =  log  1  -  log  425 
4^o 

=  0  -  2.6284 

But  since  we  always  wish  to  have  the  mantissa  of  a  logarithm 
positive,  we  write  0  =  10  —  10,  and  subtract  2.6284  from  this, 

as  follows: 

log  1  =  10.0000  -  10 

log  425  =    2.6284 

colog  425  =    7.3716  -  10. 


IV,  §72]  LOGARITHMS  85 

In  practice  this  is  done  mentally  by  beginning  at  the  left  not 
omitting  the  characteristic,  and  subtracting  each  digit  from  9, 
except  the  last  significant  digit,  which  is  subtracted  from  10. 

In  the  process  of  division  subtracting  the  logarithm  of  a 
number  and  adding  its  cologarithm  are  equivalent  operations 
since  dividing  by  N  is  equivalent  to  multiplying  by  its  reciprocal. 

72.  Computation  by  Logarithms.  It  should  be  kept  in 
mind  that  a  logarithm  is  unchanged  if  at  the  same  time  any 
given  number  is  added  to  and  subtracted  from  it.  This  is  useful 
in  two  cases: 

First.  When  we  wish  to  subtract  a  larger  logarithm  from  a 
smaller; 

Second.     When  we  wish  to  divide  a  logarithm  by  an  integer. 

EXAMPLE  1.     Find  the  value  of  27.4  -f-  652. 

log  27.4=    1.4378 

=  11.4378  -  10 
log  652  =    2.8142 


log  x  =    8.6236  -  10 
x  =    0.04304 

EXAMPLE  2.     Find  the  value  of  (0.0773)1/3. 

log  0.0773  =  8.8882  -  10. 

It  is  convenient  to  have,  after  division  by  3,  —  10  after  the  mantissa; 
hence,  before  dividing  we  add  20.0000  -  20. 

log  0.0773  =  28.8882  -  30  (divide  by  3), 
log  *  =    9.6294  -  10 
x  =    0.4250 

EXAMPLE  3.     Find  the  value  of  '  (42>6) 


["  (42.6)  (- 3.14)  I' 

02.4 


We  have  no  logarithms  of  negative  numbers,  but  an  inspection  of 
this  problem  shows  that  the  result  will  be  negative  and  numerically 


86  MATHEMATICS  [IV,  §72 

the  same  as  though  all  the  factors  were  positive;  hence  we  proceed  as 
follows: 

log  42.6    =  1.6294 
log  3.14  =  0.4969 
colog  62.4     =  8.2048  -  10  (add) 

3)0.3311          (divide  by  3) 

log  (-  x)  =  0.1104 

-  x  =  1.290,  whence  x  =  -  1.290. 

EXERCISES 

Find  approximate  values  of  the  following  by  aid  of  logarithms. 
1.  231.6  X  .0036.  2.  79  X  470  X  0.982. 

3.  13750  X  8799000.  4.  (-  9503)  X  (-  0.008657). 

5.  0.0356  X  (-  0.00049).  6.  9.238  X  0.9152 


8075                        .   « 

0.00542 

n 

24617 

364.9' 

0.04708  ' 

'   -0.00054' 

10. 

67  X  9  X  0.462 

11. 

9097  X  5.408 

.       12.  (2.38S)5. 

0.643  X  7095 

-  225  X  593 

X  0.8665 

13. 

(0.57)~4. 

14. 

(19/11)8. 

15. 

(1.014)25. 

16. 

A/67.54. 

17. 

A/-  0.3089. 

18. 

GV(  -  9.718)3. 

19. 

85/4. 

20. 

(0.001  )2'3. 

21. 

(29^9r)3/2. 

22. 

(6f)3-4. 

23. 

(-  9306)3/7. 

24. 

(0.0067)2-5. 

25. 

vixVif. 

26. 

Vol 

27. 

(0.  00068)  ~6/4. 

(0.009)3/5  ' 

00 

/  854  X  A/0.042 

!                         on    3  |7"4  X  92"«  X  (0.01  )i/» 

A/OOOl '  *    (0.00026) s  X  59681/3 

30.    V6A/0.5804  A/0.2405.  31.  (6.89  X  lO"22)16/17. 

Ans.  1.21  X  10-20. 

32.  (5.67  X  10-18)9/11.  33.  8.4 

M[(4.5  X  lO-^lO6-58]1 

Ans.  7.76  X  lO"6.  Ans.  5.51  X  107. 

34.  The  amount  a  of  a  principal  p  at  compound  interest  of  rate  r 
for  n  years  is  given  by  the  formula:  a  =  p(l  +  r)n.  Find  the  amount 
of  $486  in  5  years  at  five  per  cent,  (r  =  .05)  if  the  interest  is  com- 
pounded annually.  Ans.  $620.27 


IV,  §72]  LOGARITHMS  87 

35.  Find  the  amount  of  $384  in  40  years  at  four  per  cent.,  interest 
compounded  annually.  ,  Ans.   $1,843.59. 

36.  Find  the  simple  interest  on  $6,237.43  for  7  years  at  six  per  cent. 
Would  the  computation  made  with  four-place  logarithms,  be  sufficiently 
accurate  for  commercial  purposes?     Explain.  Ans.   $2619.72. 

37.  The  weight  P  in  pounds  which  will  crush  a  solid  cylindrical  cast- 
iron  column  is  given  by  the  formula 

,73.56 

P  =  98920^,, 

where  d  is  the  diameter  in  inches  and  I  the  length  in  feet.  What  weight 
will  crush  a  cast-iron  column  6  feet  long  and  4.3  inches  in  diameter? 

[RiBTZ  AND  CRATHORNE]  Ans.  834,200  Ibs. 

The  area  A  in  acres,  of  a  triangular  piece  of  ground,  whose  sides  are 
a,  b,  c,  rods,  is  given  by  the  formula 

_  Vs(s  —  a)(s  —  b)(s  —  c) 
160 

where  s  =  %(a  +  6  +  c).  Compute  the  areas,  in  acres,  of  the  follow- 
ing triangles: 


38.  a  =  127.6, 
39.  a  =      0.9, 
40.  a  =  408, 
41.  a  =    63.89, 

b  =  183.7, 
b  =      1.2, 
b  =    41, 
6  =  138.24, 

c  =  201.3. 
c  =       1.5. 
c  =  401. 
c  =  121.15. 

42.  The  percentage  earning  power,  E,  of  an  individual,  in  so  far  as 
it  depends  upon  the  eyes  is  given  by  Magnus  by  the  formula 


E  =  c 


where  x  takes  one  of  the  values  5,  7,  or  10,  C  being  the  maximal  central 
visual  acuity,  VPi  the  visual  field,  A/Af  the  action  of  the  extrinsic 
muscles,  Cj  and  C2  the  central  visual  acuity  of  each  eye,  and  A/P2  the 
peripheric  vision.  Compute  the  value  of  #  if  C  =  1,  PI  =  1,  M  =  1, 
Ci  =  1,  Cs  =  0.58,  x  =  10,  P2  =  1.  Ans.  97.2% 

43.  Compute  E  if  d  =  0.41,  C2  =  0.25,  x  =  5,  P2  =  M  =  Pi  =  1, 
C  =  0.41.  Ans.  33.06%. 


MATHEMATICS 


[IV,   §72 


44.  The  percentage  earning  ability  E,  as  dependent  upon  the  eyes 
is  given  by  Magnus  as 

E  = 


where  F  —  functional  ability,  V  =  necessary  knowledge,  K  —  the 
ability  to  compete  (demand  for  him),  x  has  one  of  the  values  5,  7,  or  10. 
Compute  E  for  F  =  0.78792,  V  =  1,  x  =  10,  K  =  0.39396. 

Am.  71.78%. 

45.  Compute  E  f  or  F  =  0.8254,  x  =  10,  K  =  0.4127,  7  =  1. 

Ana.   75.52%. 

46.  When  w  grams  of  a  substance  is  dissolved  in  v  liters  of  water  at  t° 
centigrade,  the  osmotic  pressure,  p,  of  the  solution  and  the  molecular 
weight,  M,  of  the  solute  are  connected  by  the  equation 

pv  =  0.082  (273  +  t)w/M. 

Compute  the  molecular  weight  of  cane  sugar  from  the  data 
(a)  p  =  12.06,  v  =  1,  t  =  22.62,  w  =  171.0  Ans.   343.7 

(6)  p  =  24.42,  »  =  3,J  =  23.56,  w  =  102.6  Ana.   340.5 

Compute  the  osmotic  pressure  for  glucose  solution,  given 

(c)  v  =  1,  t  =  26.90,  w  =  72,     M  =  180.21  4ns.    9.824 

(d)  v  =  2,  t  =  22,20,  to  =  360,  M  =  178.46  Ans.    24.36 

73.  The  Slide  Rule.  The  slide-rule  is  an  instrument  for 
carrying  out  mechanically  the  operations  of  multiplication  and 
division.  It  is  composed  of  two  pieces,  usually  about  the  shape 
of  an  ordinary  ruler;  one  of  the  pieces  (called  the  slide,)  fits  in 
a  groove  in  the  other  piece.  Each  piece  is  marked  in  divisions 


FIG.  22 


(Fig.  22),  such  that  the  distance  from  one  end  (e.  g.,  A)  is  equal 
to  the  logarithm  of  the  number  marked  on  it. 

To  multiply  one  number  (e.  g.,  2.5)  by  another  (e.  g.,  2)  we 


II,  §73] 


LOGARITHMS 


89 


set  the  point  marked  1  on  scale  B  opposite  the  point  marked  2.5 
on  scale  A  (see  Fig.  23).     Then  the  product  appears  on  scale  A 


FIG.  23 

opposite  the  point  2  on  scale  B.  Thus  5  on  scale  A  lies  oppo- 
site 2  on  scale  B  in  Fig.  23.  This  follows  from  the  fact  that 
log  2.5  +  log  2  =  log  5. 

Likewise,  if  1  on  scale  B  is  set  opposite  any  number  a  on 
scale  A,  then  we  find  opposite  any  number  6  on  scale  B  the 
number  ab  on  scale  A. 

Divisions  can  be  performed  by  reversing  this  process.  Thus 
if  6  on  scale  B  be  set  opposite  c  on  scale  A,  the  1  on  scale  B 
will  be  opposite  c/b  on  scale  A. 

A  little  practice  with  such  a  slide-rule  will  make  clear  the 
actual  procedure  in  any  case. 

Scales  C  and  D  are  made  just  twice  the  size  of  scales  A  and  B. 
It  follows  that  any  number  on  scale  C,  for  example,  is  exactly 
opposite  the  square  of  that  number  on  scale  A.  This  facilitates 
the  finding  of  squares  and  square  roots,  approximately. 

Scales  C  and  D  may  be  used  in  place  of  scales  A  and  B  for 
multiplication  and  division.  Indeed,  after  some  practice, 
scales  C  and  D  will  be  preferred  for  this  purpose. 

More  elaborate  slide-rules,  marked  with  several  other  scales, 
are  for  sale  by  all  supply  stores.  Descriptions  of  these  and 
full  directions  for  their  use  will  be  found  in  special  catalogs 
issued  bv  instrument  makers. 


90  MATHEMATICS  [IV,  §73 

A  simple  slide-rule  can  be  bought  at  a  moderate  price.  One 
sufficient  for  temporary  practice  may  be  made  by  the  student 
by  cutting  out  the  large  figure  printed  on  one  of  the  fly-leaves 
of  this  book,  and  following  the  directions  printed  there. 

The  student  should  secure  some  form  of  slide-rule  and  he 
should  use  it  principally  in  checking  answers  found  by  other 
processes. 

As  exercises  the  teacher  may  assign  first  very  simple  products 
and  quotients.  When  the  operation  of  the  slide-rule  has  been 
mastered,  the  student  may  check  the  answers  to  the  exercises 
on  p.  86. 


CHAPTER  V 


TRIGONOMETRY 

74.  Introduction.     The  sides  and  angles  of  a  plane  triangle 
are  so  related  that  any  three  given  parts,  provided  at  least  one 
of  them  is  a  side,  determine  the  shape  and  the  size  of  the  triangle. 

Geometry  shows  how,  from  three  such  parts,  to  construct  the 
triangle. 

Trigonometry  shows  how  to  compute  the  unknown  parts  of  a 
triangle  from  the  numerical  values  of  the  given  parts. 

Geometry  shows  in  a  general  way  that  the  sides  and  angles 
of  a  triangle  are  mutually  dependent.  Trigonometry  begins 
by  showing  the  exact  nature  of  this  dependence  in  the  right 
triangle,  and  for  this  purpose  employs  the  ratios  of  the  sides. 

75.  Definitions  of  Trigonometric  Functions.    Let  A  be  any 
acute  angle.     Place  it  on  a  pair  of  axes  as  in  Fig.  24,  with  the 
vertex  at  the  origin,  one  side  along 

the  ar-axis  to  the  right,  and  the 
other  side  in  the  first  quadrant. 
On  this  side  choose  any  point  M  (ex- 
cept 0)  and  drop  M  N  perpendic- 
ular to  the  or-axis.  Let  OM  =  r; 
then  by  plane  geometry, 


x- 


FIG.  24 


r  =  v  ar* 

where  x  and  y  are  the  coordinates  of  the  point  M.  The  differ- 
ent ratios  of  pairs  of  the  three  numbers  x,  y,  and  r,  are  designated 
as  follows 

y  _  ordinate  _ 

r        radius 

x      abscissa 


(1) 
(2) 
(3) 


r        radius 
y  _  ordinate  _ 
x      abscissa 


of  angle  A,  written  sin  A, 
=  the  cosine  of  angle  A,  written  cos  A, 


tangent  of  angle 
91 


>  written  tan  A. 


92 


MATHEMATICS 


[V,  §  75 


The  reciprocals  of  these  rations  are  also  used, 

,.,    x      abscissa 

(4)   ~  = : 

y      ordinate 


=  the  cotangent  of  angle  A,  written  ctn  A, 


(5)  -  =  —          -  =  secant  of  angle  A,  written  sec  A, 
x      abscissa 

(6)  -  =  —          -  =  cosecant  of  angle  A,  written  esc  A. 
y      ordinate 

These  six  ratios  are  called  the  trigonometric  functions  of  the 
angle  A.  They  do  not  at  all  depend  upon  the  choice  of  the 
point  M  on  the  side  of  the  angle  but  only  upon  the  magnitude 
of  the  angle  itself. 

For  if  we  choose  any  two  points  M'  and  M "  on  the  side  of  the 


. 

_ 

' 

t 

*.! 

/ 

-/' 

/ 

„ 

/ 

, 

/ 

M 

/ 

s 

/ 

/ 

> 

•\j 

i 

y 

M 

/ 

y 

' 

^" 

•^ 

/ 

/ 

^ 

x 

<" 

>• 

y 

j 

/ 

K 

B 

. 

^ 

** 

V 

^ 

/ 

3 

„ 

/ 

\ 

^ 

s 

\ 

o 

,/ 

X 

A 

A 

0 

^ 

^ 

~x 

V 

A' 

\ 

X 

' 

X 

FIG.  25 

same  angle  A,  and  denote  their  coordinates  by  (xf,  y'}  and  (x", 
y")  respectively,  then  by  similar  triangles, 


~.  =  ^--   =  sin  A, 


--  =  %—  =  tan  A,  etc. 


But  if  we  take  two  points  M'  and  M"  at  the  same  distance 
r  from  0  on  the  sides  of  two  different  angles  A  and  B,  then 

y'      v" 

sin  A  =  '—  ^  '-*—  =  sin  B, 
r         r 

tan  A  =^-^^-  =tanB, 
x       x 

and  similarly  the  other  functions  of  A  and  B  are  unequal. 


V,  §  76] 


TRIGONOMETRY 


93 


From  these  definitions  we  deduce  the 
following  relations  which  are  of  fundamen- 
tal importance  in  computing  the  unknown 
parts  of  right  triangles. 

In  any  right  triangle,  having  fixed  atten- 
tion on  one  of  the  acute  angles, 


side  adjacent 
FIG.  26 


(7) 
(8) 

(9) 


The  side  opposite  =  hypotenuse  X  sine. 

also  =  side  adjacent  X  tangent. 
The  side  adjacent  =  hypotenuse  X  cosine. 

also  =  side  opposite  X  cotangent. 

side  opposite 
The  hypotenuse  = 


also  = 


sine 

side  adjacent 

cosine 


EXERCISES 

Find  the  six  functions  of  each  of  the  acute  angles  in  the  right  tri- 
angle whose  sides  are : 

1.   3,  4,  5.  2.   9,  40,  41.  3.   60,  91,  109. 

4.    7,  24,  25.  5.    16,  63,  65.  6.    20,  99,  101. 

7.   20,  21,  29.  8.   36,  77,  85.  9.    12,  35,  37. 

10.    2n  +  1,  2n(n  +  1),  2n2  +  2n  +  1.     11.  2n,  n2  -  1,  n2  +  1. 
12.   2(n  +  1),  n(n  +  2),  n2  +  2n  +  2.      13.  a(62  -  c2),  2abc,  a(62  +  c2). 

76.  Functions  of  Complementary  Angles.    Let  A  and  B  be 
the  acute  angles  in  any  right  triangle.     Then, 

?/  *T 

B      sin  A  =  cos  B  =  - ,    cos  A  =  sin  B  =  — , 
r  r 

tan  A  =  ctn  #  =  -,    ctn  A  =  tan  B  =  - 

x  y 


x 
FIG.  27 


sec  A  =  esc  B  =  — ,    esc  A  =  sec  B  =  - 
x  y 


94 


MATHEMATICS 


[V,  §76 


Since  A  -\-  C  =  90°  (i.  e.,  A  and  C  are  complementary),  the 
above  results  may  be  stated  in  compact  form  as  follows: 

A  function  of  an  acute  angle  is  equal  to  the  co-function  of  its 
complementary  acute  angle. 

77.  Functions  of  30°,  45°,  60°.  On  the  sides  of  a  right 
angle  lay  off  unit  distances  A B  and  AC  and  draw  BC,  forming 
an  isosceles  right  triangle,  Fig.  28.  The  angles  at  B  and  C  are 
each  45°,  and  the  hypotenuse  BC  is  equal  to  V2~  (why?). 


FIG.  28 


\ 


AID  B 

FIG.  29 


From  the  definitions, 

sin  45°  =  cos  45°  =  I/ A/2  =   A/2/2. 
tan  45°  =  ctn  45°  =  1. 
sec  45°  =  esc  45°  =   A/2/1  =   A/2. 

Construct  an  equilateral  triangle  whose  sides  are  2  units  long, 
Fig.  29.  Bisect  one  of  its  angles  forming  a  right  triangle  ACD, 
in  which  A  =  60°,  C  =  30°,  and  the  altitude  CD  is  equal  to  A/3 
(why?).  Then  from  the  definitions, 


sin  60°  =  cos  30°  =   A/3/2. 
cos  60°  =  sin  30°  =  1/2. 
tan  60°  =  ctn  30°  =   A/3. 


ctn  60°  =  tan  30°  =  l/A/3. 
sec  60°  =  esc  30°  =  2. 
esc  60°  =  sec  30°  =  2/  A/3. 


V,  §  78] 


TRIGONOMETRY 


78.  Eight  Fundamental  Relations.     The  following  relations 
hold  for  the  trigonometric  functions  of  any  acute  angle  A, 

(10)  sin  A  esc  A  =  1,  sine  and  cosecant  are  reciprocals ; 

(11)  cos  A  sec  A  =  1,  cosine  and  secant  are  reciprocals ; 

(12)  tan  A  ctn  A  =  1,  tangent  and  cotangent  are  reciprorocals ; 


(13)    tan  A  = 


sin  A 


cos  A 

(15)  sin2 A  +  cos2 A  =  1 ; 

(16)  tan2 A  +  1  =  sec2  A  • 


(14)    ctn  A  = 


cos  A 
sin  A  ' 


(17)   ctn2 A  +  I  =  esc2 A. 


These  eight  identities  are  fundamental  relations  and  should 
be  thoroughly  learned  by  the  student. 

They  may  be  proved  as  follows:  (10),  (11),  (12)  are  direct 
consequences  of  the  definitions  in  §  75.  To  prove  (13),  we  have 

tan  A  =  - ,  sin  A  =  - ,  cos  A  =  - , 
x  r  r 

whence 


sin  A      11      x     11 

-=--r--  =  -=  tan  A. 
cos  A      r      r      x 


FIG.  30 


Similarly, 

cos  A      x  .  y     x 

-  =  -  -:-  -  =  -  =  ctn  A. 
sin  A      r      r      y 

From  Fig.  30, 

(18)  x2  +  y2  =  r2. 

Dividing  through  by  r2,  we  have 

^+^-1, 


whence       cos2  A  +  sin2  A  =  1. 

Similarly,  dividing  (18)  through  by  x2,  and  then  by  yz  we  prove 

(16)  and  (17). 

If  the  value  of  one  function  of  an  angle  is  known,  the  values 
of  all  the  others  can  be  found  by  means  of  these  relations. 


96  MATHEMATICS  [V,  §78 

EXAMPLE.     Given  sin  A  =  1/2.     Then, 

cos  A  =  Vl  —  sin2  A  =  A/I  =  f  V3, 
and,  by  (13), 

tan  A  =  1/V3  =  iV§. 

Since  the  other  three  functions  are  reciprocals  of  these  three,  we  have 
ctn  A  =  V3,        sec  A  =  f  V§,        esc  A  =  2. 

The  values  of  these  functions  can  also  be  found  graphically  by  con- 
structing a  right  triangle  the  ratio  of  whose 
sides  are  such  as  to  make  the  sine  of  one  angle 
equal  to  1/2.  This  can  evidently  be  done 
by  making  the  side  opposite  equal  to  1  and 
the  hypotenuse  equal  to  2;  then  the  side  ad- 
jacent is  equal  to  >/3.  (Why?)  The  other 
functions  can  now  be  read  directly  from  the 

figure,  using  the  definitions.     Thus,  tan  A  =  side  opposite  -f-  side  ad- 
jacent =  l/A/3  =  f  A/3. 

EXERCISES 

1.  Given  sin  40°  =  cos  50°;  express  the  other  functions  of  40°  in 
terms  of  functions  of  50°. 

2.  The  angles  45°  +  A  and  45°  —  A  are  complementary;  express 
the  functions  of  45°  +  A  in  terms  of  the  functions  of  45°  —  A. 

3.  A   and  90°  —  A   are  complementary;  express  the  functions  of 
90°  —  A  in  terms  of  the  functions  of  A. 

4.  Construct  a  right  triangle,  having  given 

(a)  hypotenuse  =  6,  tangent  of  one  angle  =  3/2. 

(b)  cosine  of  one  angle  =  1/2,  side  opposite  =  3.5. 

(c)  sine  of  one  angle  =  0.6,  side  adjacent  =  2. 

(d)  cosecant  of  one  angle  =  4,  side  adjacent  =  4. 

(e)  one  angle  =  45°,  side  adjacent  =  20. 
(/)  one  angle  =  30°,  side  opposite  =  25. 

5.  In  Ex.  4,  compute  the  remaining  parts  of  each  triangle. 

6.  Express  each  of  the  following  as  a  function  of  the  complementary 
angle: 

(a)  sin  30°.  (6)  tan  89°.  (c)   esc  18°  10'.      (d)  ctn  82°  19'. 

(e)   cos  45°.  (/)ctn!5°.  (g)  cos  37°  24'.      (A)  esc  54°  46'. 


V,  §80]  TRIGONOMETRY  97 

7.  Express  each  of  the  following  as  a  function  of  an  angle  less 
than  45°: 

(a)  sin  60°.  (6)   tan  57°.  (c)   esc  69°  2'.        (d)  ctn  89°  59'. 

(e)   cos  75°.  (/)ctn84°.  (g)  cos  85°  39'.      (ft)  esc  45°  13'. 

8.  Prove  that  if  A  is  any  acute  angle, 

(a)  sin  A -sec  A  =  tan  A.  (6)    sin  A- ctn  A  =  cos  A. 

(c)  cos  A -esc  A  =  ctn  A.  (d)  tan  A-  cos  A  —  sin  A. 

(e)  sin  A-sec  A-ctn  A  =  1.  (/)  cos  A-csc  A-tan  A  =  1. 

(g)  (sin  A  +  cos  A)2  —  1  +  2  sin  A  cos  A, 

(h)  (sec  A  +  tan  A) (sec  A  —  tan  A)  =  1. 

(i)  (1  +  tan2  A)  sin2  A  =  tan2  A. 

0')  (1  -  sin2  A)  esc2  A  =  ctn2  A. 

(k)  sin4  A  —  cos4  A  =  sin2  A  —  cos2  A. 

(0  tan2  A  cos2  A  +  cos2  A  =  1. 

(m)  (sin  A  +  cos  A)2  +  (sin  A  —  cos  A)2  =  2. 

(n)  sec  A  —  cos  A  =  sin  A  tan  A. 

(o)  (sin2  A  —  cos2  A)2  =  1  —  4  sin2  A  cos2  A. 

(p)  (1  -  tan2  A)2  =  sec4  A  -  4  tan2  A. 

9.  Express  the  values  of  all  the  other  functions  of  A  hi  terms  of 
(a)  sin  A,    (6)  cos  A,    (c)  tan  A,     (d)  ctn  A,    (e)  sec  A,     (/)  esc  A. 

79.  Solution  of  Right  Triangles.     The  values  of  the  six 
trigonometric  ratios  have  been  computed  for  all  acute  angles, 
and  recorded  in  convenient  tables.     They  are  given  to  four 
decimal  places  in  Table  II,  at  the  end  of  the  book.     These 
tables,  together  with  the  definitions  of  the  functions,  enable 
us  to  solve  all  cases  of  right  triangles. 

80.  General  Directions  for  Solving  Right  Triangles. 

(1)  Draw  a  diagram  approximately  to  scale,  indicating  the 
given  parts.     Mark  the  unknown  parts  by  suitable  letters,  and 
estimate  their  values. 

(2)  //  one  of  the  given  parts  is  an  acute  angle,  consider  the 
relation  of  the  known  parts  to  the  one  which  it  is  desired  to  find, 
and  apply  the  appropriate  one  of  formulas  (7),  (8),  (9),  p.  93. 

(3)  //  two  sides  are  given,  and  one  of  the  angles  is  desired, 


98 


MATHEMATICS 


[V,  §  80 


think  of  the  definition  of  that  function  of  the  angle  which  em- 
ploys the  two  given  sides. 

(4)  Check  the  results.  The  larger  side  must  be  opposite  the 
larger  angle,  and  the  square  of  the  hypotenuse  must  be  equal  to 
the  sum  of  the  squares  of  the  other  two  sides. 

The  following  examples  illustrate  the  process  of  solution. 

EXAMPLE  1.  Given  the  hypotenuse  =  26,  and 
one  angle  =  43°  17';  find  the  two  sides  and  the  other 
acute  angle.  Do  not  use  logarithms. 

Draw  a  figure  ABC  in  which  AC  =  26,  A  =  43° 
17'  and  denote  the  unknown  parts  by  suitable  let- 
ters, x,  y,  and  C.  Find  C  as  the  complement  of 

A: 

90°  00' 
A  =  43°  17' 
C  =  56°  43' 


26 


x  B 

FIG.  32 


To  find  x  note  that  it  is  adjacent  to  the  given  angle  and  that  the  hypo- 
tenuse is  given, 
Then  by  (8)  §  75 

x  =  26  cos  43°  17' 
cos  43°  17'  =    0.7280 
26 


Similarly  by  (7)  §75 

y  =  26  sin  43°  17' 


sin  43°  17'  =    0.6856 
26 


4368 
1456 
x  =  18.928 

CHECK:  tan  A  =  y/x  =  0.9418. 

EXAMPLE  2.     An  acute  angle 
10'  and  the  opposite  side  is  78. 
Solve  by  means  of  logarithms. 
By  (8)  §  75 

x  =  78  ctn  62°  10' 

log  78  =  11.8921  -  10 
log  ctn  62°  10'  =    9.7226  -  10 
log  x  =     1.6147 
x  =  41.18 


41136 
13712 
y  =  17.8256 

tan  43°  17'  =  0.9418. 

of  a  right  triangle  is  62° 
Find  the  other  parts. 

By  (9)  §  75 

r  =  78/sin  62°  10' 

log  78  =  11.8921  -  10 
log  sin  62°  10'  =    9.9466  -  10 
log  r  =    1.9455 
r  =  88.20 


V,  §81] 


TRIGONOMETRY 


99 


CHECK: 


r  =    88.20 

x  =    41.18 

r  +  x  =  129.38 

r  -  x  =    47.02 


log  (r  +  x)  =  2.1119 

log  (r  -  x)  =  1.6723 

3.7842 

log  782  =  3.7842 


EXAMPLE  3.  The  hypotenuse  of  a  right  triangle  is  42.7  and  one 
side  is  18.5.  Find  the  other  parts.  To  find  one  of  the  angles,  as  C, 
note  that  the  hypotenuse  and  side  adjacent  are  known.  Then 


FIG.  34 


ICC 

cos  C  =    ~  =  0.4332 


C  =  64°  19'.6 


42?7    =  1823.29 
18.52  =    342.25 


x2  =  1481.04 
x  =      38.48 


SOLUTION  BY  LOGARITHMS. 

18.5 
cosC  =— . 

log  18.5  =  1.2672 
log  42.7  =  1.6304_ 
log  cos  C  =  9.6368 
C  =  64°  19' 


A  =  90°  -  C  =  25°  40'.4 

CHECK:  x  =  18.5  ctn  25°  40'.4 

=  18.5  X  2.0803  =  38.48 


=  42.72  -  18.52 

=  61.2  X  24.2 

log  24.2  =    1.3838 

log  61. 2  =  1.7868 
log  x2  =  3.1706 
log  x  =  1.5853 
x  =  38.48 


81.  Graphical  Solution.  As  shown  in  §  35,  if  the  triangle  be 
drawn  to  scale,  the  unknown  sides  can  be  read  off  on  the  scale, 
and  the  unknown  angles  on  a  protractor.  The  results  so  ob- 
tained will  be  accurate  enough  to  detect  any  large  errors  in  the 
computations. 


100 


MATHEMATICS 


[V,  §81 


EXERCISES 

Let  A,  B,  C  represent  the  three  angles  of  any  triangle  and  a,  b,  c 
the  sides  opposite  these  angles. 

1.  Solve  graphically  the  following  triangles: 

(a)  a  =  5,  b  =  4,  c  =  7.    Ans.  A  =  44°  30',  B  =  34°,  C  =  101°  30'. 
Ans.  A  =  22°,         B  =  60°,  C  =  98°. 
Ans.  A  =  38°,         B  =  60°,  C  =  82°. 


(6)  a  =  3,  b  =  7,  c  = 

(c)  a  =  5,  b  =  7,  c  = 

(d)  a  =  8,  b  =  7,  B  =  60°. 

Ans.  Ai  =  82°,  A2  =  98 

(e)  a  =  3,  b  =  5,  c  =  7. 
(/•)  a  =  7,  A  =  120°,  b  =  5. 
(0)   o  =  42,  b  =  51,  A  =  55°. 

Ans.  Bi  =  84°,  B2  =  96°, 


Ci  =  38°,  C2  =  22°,  ci  =  5,  c2  =  3. 
Ans.  4  =  22°,  B  =  38°,  C  =  120°. 
Ans.  £  =  38°,  C  =  22°,  c  =  3. 


=  41°,  C2  =  29° 
2.  Solve  the  following  right  triangles  (C  =  90°). 


=  34,  c2  =  25. 


Required  parts  (Answers). 
B  =  60°, 
B  =  45°, 
B  =  30°, 
A  =  65°, 
A  =  50°, 
A  =  20°, 
A  =  45°, 
A  =  36°  52' 
A  =  4°  46', 
B  =  67° 
5  =  53°, 
A  =  48°, 

3.  The  width  of  the  gable  of  a  building  is  32  ft.  9  in.     The  height  of 
the  ridge  of  the  roof  above  the  plates  is  14  ft.  6  in.     Find  the  inclina- 
tion of  the  roof,  and  the  length  of  the  rafters. 

Ans.    41°  32',  21  ft.  10  in. 

4.  The  steps  of  a  stairway  have  a  tread  of  10  in.  and  a  rise  of  7  in. ; 
at  what  angle  is  the  stairway  inclined  to  the  floor?  Ans.    35°. 

5.  The  shadow  of  a  tower  200  ft.  high  is  252.5  ft.  long.     What  is  the 
angle  of  elevation  of  the  sun? 


Given  parts. 

(a) 

A 

=  30°, 

a 

=  12, 

(6) 

A 

=  45°, 

b 

=  8, 

(c) 

A 

=  60°, 

c 

=  20, 

(d) 

B 

=  25°, 

a 

-72, 

(^ 

B 

=  40°, 

b 

=  33, 

(!) 

B 

=  70°, 

c 

=  81, 

(?) 

a 

=  6, 

b 

=  6, 

(A) 

a 

=  3, 

c 

=  5, 

(*) 

b 

=  12, 

c 

=  13, 

0') 

A 

=  23°, 

a 

=  3.246, 

(fc) 

A 

=  37°, 

b 

=  7.28, 

(0 

B 

=  42°, 

c 

=  1021, 

b 

=  20.78, 

c 

=  24 

a 

=  8, 

c 

=  11.31 

a 

=  17.32, 

6 

=  10 

b 

=  33.57, 

c 

=  79.44 

a 

=  39.33, 

c 

=  51.34 

a 

=  27.70, 

b 

=  76.12 

B 

=  45°, 

c 

=  8.484 

B 

=  53°  8', 

b 

=  4 

B 

=  85°  14', 

a 

=  i 

b 

=  7.647, 

c 

=  8.307 

a 

=  5.486, 

c 

=  9.116 

a 

=  758.7, 

b 

=  713.8 

V,  §82]  TRIGONOMETRY  101 

6.  A  cord  is  stretched  around  two  wheels  with  radii  of  7  feet  and  1 
foot  respectively,   and  with  their  centers   12  feet  apart.     Find   the 
length  of  the  cord.  Ans.  12  V3  +  10w  =  52.2  ft. 

7.  Two  objects  A,  B  in  a  rectangular  field  are  separated  by  a  thicket. 
To  determine  the  distance  between  them,  the  lines  AC  =  45  rods, 
BC  =  36  rods,  are  measured  parallel  to  the  sides  of  the  field.     Find 
the  distance  AB.  Ans.  57.63 

8.  One  bank  of  a  river  is  a  bluff  rising  75  ft.  vertically  above  the 
water.     The  angle  of  depression  of  the  water's  edge  on  the  opposite 
bank  is  20°  27'.     Find  the  width  of  the  river.  Ans.  201.1 

9.  A  smokestack  is  secured  by  wires  running  from  points  on  the 
ground  35  ft.  from  its  base  to  points  3  ft.  from  its  top.     These  wires 
are  inclined  at  an  angle  of  40°  to  the  ground,     (a)  What  is  the  height 
of  the  smokestack?     (6)  The  length  of  the  wires?     (c)  What  is  the 
least  number  of  wires  necessary  to  secure  the  stack?     If  they  are  sym- 
metrically placed,  how  far  apart  are  their  ground  ends?     (d)  How 
far  are  the  lines  joining  their  ground  ends  from  the  foot  of  the  stack? 
(e)  From  the  top  of  the  stack?     (/)  What  angle  do  the  wires  make 
with  these  lines?     (<?)  With  each  other?     (h)  What  angle  does  the 
plane  of  two  wires  make  with  the  ground?     (i)  What  angle  does  the 
perpendicular  from  the  foot  of  the  stack  on  this  plane  make  with  the 
ground?     (j)  What  is  its  length?  [DURFEE] 

10.  A  tree  stands  on  a  horizontal  plane.     At  one  point  in  this  plane 
the  angle  of  elevation  of  the  top  of  the  tree  is  30°,  at  another  point 
100  feet  nearer  the  base  of  the  tree  the  angle  of  elevation  of  the  top  is  45°. 
Find  the  height  of  the  tree. 

11.  Find  the  length  of  a  ladder  required  to  reach  the  top  of  a  building 
50  ft.  high  from  a  point  20  ft.  in  front  of  the  building.     What  angle 
would  the  ladder  in  this  position  make  with  the  ground? 

82.  General  Angles.  Rotation.  Up  to  this  point  we  have 
defined  and  used  the  trigonometric  functions  of  acute  angles 
only.  Many  problems  require  the  consideration  of  obtuse  angles 
and  others,  particularly  those  concerned  with  the  rotating  parts 
of  machinery,  involve  angles  greater  than  180°  or  360°  even,  and 
it  is  necessary  to  distinguish  between  parts  in  the  same  or  par- 
allel planes  which  rotate  in  the  same  or  in  opposite  directions. 


102 


MATHEMATICS 


[V,  §82 


An  angle  may  be  thought  of  as  being  generated  by  the  rota- 
tion of  one  of  its  sides  about  the  vertex;  its  first  position  is 

called  the  initial  side,  its  final 
position  the  terminal  side  of 
the  angle.  An  angle  gener- 
ated by  rotation  opposite  to 
the  motion  of  the  hands  of 
a  clock  (counterclockwise)  is 
FIG.  35  said  to  be  positive;  an  angle 

generated  by  clockwise  rotation  is  said  to  be  negative.  In  draw- 
ings a  curved  arrow  may  be  used  to  show  the  direction  of  rota- 
tion, the  arrow  head  indicating  the  terminal  side. 

83.  Trigonometric  Functions  of  any  Angle.     Let  $  =  XOP 

be  any  angle  placed  with  its  vertex  at  the  origin  and  its  initial 

side  along  the  positive  z-axis.     Let  P  be  any  point  (except  0) 

kY 


J"        a> 


FIG.  36 

on  the  terminal  side  and  let  x,  y  be  its  coordinates  (positive, 
negative,  or  zero  depending  upon  the  position  of  P  in  the  plane) ; 
let  r  be  the  distance  from  0  to  P  (always  positive).  Then  the 
trigonometric  functions  of  $  are  defined  as  follows: 


(19) 


sin  (f>  =  -  , 


cos  0  =  - . 


The  definitions  (19)  apply  to  all  angles  without  exception. 


(20) 


tan  </>  = 


sec  d>  =  -  . 
x 


V,  §83] 


TRIGONOMETRY 


103 


The  definitions  (20)  apply  to  all  angles  except  odd  multiples  of  a 
right  angle;  this  exception  is  necessary  because  for  all  such 
angles  x  is  zero. 


(21) 


ctn  <A   =  - 

v 


esc  <6  =  —  . 

y 


The  definitions  (21)  apply  to  all  angles  except  even  multiples 
of  a  right  angle;  for  all  such  angles  y  is  zero. 

These  definitions  apply  of  course  to  all  acute  angles  and 
give  the  same  values  as  the  definitions  in  §  75.  These  new 
definitions  are  more  general  because  they  apply  to  angles  to 
which  the  former  do  not  apply. 

These  ratios  are  independent  of  the  choice  of  P  on  the  terminal 
side  of  the  given  angle.  They  depend  upon  the  magnitude  and 
sign  of  the  angle.  For,  if  we  choose  a  different  point  P'  on  the 
terminal  side  of  <£,  we  shall  have 


in  magnitude  and  sign  and  this  implies  that 


"-.  =  2-i          etc. 


The  signs  of  the  trigonometric  functions  of  an  angle  0  depend 
upon  the  quadrant  of  the  plane  in  which 
the  terminal  side  of  <f>  falls  when  it  is  placed 
on  the  axes.  An  angle  <£  is  said  to  be  an 
angle  in  the  first  quadrant  when  its  ter- 
minal side  falls  in  that  quadrant,  and  simi- 
larly for  the  second,  third,  and  fourth 
quadrants.  The  signs  of  the  sine  and  the 
cosine  of  an  angle  in  each  of  the  quadrants 
should  be  thoroughly  learned.  The  accompanying  diagram  in- 
dicates these  signs. 


FIG.  37 


104 


MATHEMATICS 


[V,  §83 


The  signs  of  the  other  functions  are  determined  by  noting  that 
tan  <£  is  positive  when  sin  <£  and  cos  <j>  have  like  signs  and 
negative  when  they  have  unlike  signs;  and  that  reciprocals 
have  like  signs. 

84.  The    Fundamental    Rela- 
tions.    The   fundamental    identi- 
ties (10)  to  (18)  which  were  proved 
for  acute  angles  in  §  78  are  valid 
for    any    angle    whatever.       The 
proofs  which  are  similar  to  those 
already  given  are  left  to  the  student. 

85.  Quadrantal  Angles.  Let  P be 
a  point  on  the  terminal  side  of  an 
angle  0  at  a  distance  r  from  the  origin. 

When  (j>  =  0°,  P  coincides  with  PI  and  its  coordinates  are 
x  =  r  and  y  =  0;  then  by  §  83 


sin  0°  =  -  =  0, 

r 

tan  0°  =  -  =  0. 
x 


cos  0°  =  - 
r 


1, 


sec  0°  =  -  =  1. 
x 


The  angle  0°  has  no  cotangent  nor  cosecant. 

When  0  =  90°,  P  coincides  with  P2,  x  =  0,  y  =  r;  then 


sin  90°  =      =  1, 
r 

ctn  90°  =  -  =  0, 

y 


cos  90°  =  -  =  0, 
r 

esc  90°  =  -  =  1. 

y 


The  angle  90°  has  no  tangent  nor  secant. 

When  <j>  =  180°,  P  coincides  with  P3,  x  =  -  r,  y  =  0;  then 


sin  180°  =  -  =  0, 
r 

tan  180°  =  -  =  0, 


cos  180°  =  -  =  -  1, 
r 

sec  180°   = 


The  angle  180°  has  no  cotangent  nor  cosecant. 


V,  §86]  TRIGONOMETRY  105 

When  0  =  270°,  P  coincides  with  P4,  x  =  0,  y  =  —  r;  then 

sin  270°  =  -  =  -  1,  cos  270°  =  -  =  0, 

r  r 

ctn  270°  =  -  =  0,  esc  270°  =  -=_!. 

v  y 

The  angle  270°  has  no  tangent  nor  secant. 

Often  it  is  said  that  tan  90°  =  °o }  but  this  does  not  mean 
that  90°  has  a  tangent;  it  means  that  as  an  angle  0  increases 
from  0°  to  90°,  tan  0  increases  without  limit,  and  that  before  $ 
reaches  90°.  Similar  remarks  apply  to  the  statements  ctn  0° 
=  oo ,  tan  270°  =  oo ,  etc. 

86.  Line  Representations  of  the  Trigonometric  Func- 
tions. The  trigonometric  functions  denned  in  §  83  are  abstract 
numbers;  each  is  the  ratio  of  two  lengths.  They  are  not  lengths 
nor  lines.  They  can  however  very  conveniently  be  represented 
by  line  segments  in  the  sense  that  the  number  of  length  units  in 
the  segment  is  equal  to  the  magnitude  of  the  function,  and  the  sign 
of  the  segment  is  the  same  as  the  sign  of  the  function. 

Let  an  angle  0  of  any  magnitude  and  sign  be  placed  on  the 
axes,  Fig.  39.  With  the  origin  as  center  and  a  radius  one  unit 
length  draw  a  circle  cutting  the  positive  z-axis  at  A,  the  positive 
y-axis  at  B,  and  the  terminal  side  of  0  at  P.  Draw  tangents 
to  this  circle  at  ^4.  and  at  B  and  produce  the  terminal  side  in 
one  or  both  directions  from  0  to  cut  these  tangents  in  T  and  S 
respectively.  Draw  PQ  perpendicular  to  the  z-axis.  Then,  if 
we  agree  that  QP  shall  be  positive  upward,  OQ  shall  be  positive 
to  the  right,  and  that  OT,  or  OS,  shall  be  positive  when  it  has 
the  same  sense  as  OP  and  negative  when  it  has  the  opposite 
sense, 

QP  represents  sin  0,  OQ  represents  cos  0, 
A  T  represents  tan  0,  AS  represents  ctn  0, 
OT  represents  sec  0,  OS  represents  esc  0. 


106 


MATHEMATICS 


[V,  §86 


For,  sin  </»  =  QP/OP  =  the  number  of  units  of  length  in  QP 
since  OP  =  unit  length  and  sin  $  and  QP  agree  in  sign  from 
quadrant  to  quadrant.  Similarly  the  others  may  be  proved. 


\ 


FIG.  39 


The  student  is  warned  against  thinking  or  saying  that  "  QP 
is  the  sine  of  0  ";  say  "  The  number  of  units  in  QP  is  sin  (/>  " 
or,  "  QP  represents  sin  0." 

87.  Congruent  Angles.  Any  angle  formed  by  adding  to  or 
subtracting  from  a  given  angle  <f>,  any  multiple  of  360°  is  said 
to  be  congruent  to  0;  thus  —  217°  and  143°  are  congruent. 
It  is  obvious  from  the  definitions  and  from  the  line  representa- 
tions of  the  functions  of  an  angle  that  two  congruent  angles 
have  equal  functions.  The  functions  of  any  angle  formed  by 
adding  to  or  subtracting  from  a  given  angle  a  multiple  of  360°  are 
the  same  as  the  corresponding  functions  of  the  given  angle. 


v,  § 


TRIGONOMETRY 


107 


88.  Trigonometric  Equations.  To  solve  the  equation  sin  x 
=  1/2  is  to  find  all  angles  which  satisfy  it.  We  know  that 
x  =  30°  is  a  solution  for  sin  30°  =  1/2;  x  =  150°,  x  =  -  210°, 
x  =  750°,  are  also  solutions.  We  can  find  all  its  solutions  by 
the  following  graphical  method. 

1)  To  solve  the  equation 
sin  x  =  s. 


where  s  is  a  given  number  between 
-  1  and  +  1,  draw  a  unit  circle 
center  at  the  origin  and  on  the 
y-Sixis  lay  off  OB  =  s  (above  0  if 
s  >  0,  below  if  s  <  0)  and  through 
B  draw  a  parallel  to  the  z-axis  cut- 
ting the  circle  in  C  and  D,  Then 
the  positive  angles 


FIG.  40 


a  =  AOC        and        j8  =  AOD 

are  solutions  (and  the  only  solutions  between  0°  and  360°)  of 
the  given  equation.     Any  angle  congruent  to  a  or  to  /3  is  also  a 
solution,  and  there  are  no  others.     These  results  follow  directly 
from  the  line  representations  of  the  functions  in  §  86. 
2)  To  solve  the  equation 

cos  x  =  c, 

where  c  is  a  given  number  between  —  1  and  +  1,  draw  a  unit 
circle  center  at  the  origin,  Fig.  41,  and  lay  off  on  the  z-axis 
OB  =  c  (to  the  right  if  c  >  0,  to  the  left  if  c  <  0)  and  draw 
through  B  a  parallel  to  the  7/-axis  cutting  the  circle  in  C  and  D. 
Then  the  positive  angles 

a  =  AOC        and        0  =  AOD 

are  solutions  (and  the  only  solutions  between  0°  and  360°)  of 
the  given  equation.  Any  angle  congruent  to  a  or  to  /3  is  also  a 
solution  and  there  are  no  others. 


108 


MATHEMATICS 


[V,  §88 


FIG.  41 
3)  To  solve  the  equation 


FIG.  42 


tan  x  =  t 

where  t  is  any  given  number  whatever,  draw  a  unit  circle  center 
at  the  origin,  and  lay  off  on  the  tangent  at  A, 

AB  =  t 

and  draw  a  line  through  0  and  B  cutting  the  circle  in  C  and  D. 
Then  the  positive  angles 

a  =  AOC,        j8  -  AOD 

are  solutions  (and  the  only  solutions  between  0°  and  360°)  of 
the  given  equation.  Any  angle  congruent  to  a  or  to  /3  is  also 
a  solution,  and  there  are  no  others. 

Many  other  trigonometric  equations  can  be  reduced  to  one 
of  these  three  forms  by  the  transformations  given  in  §  78  and 
hence  can  be  solved  by  the  above  methods. 

For  example,  the  equation 


is  equivalent  to 
Again, 


tan  x  =  3. 

2  sin2  x  —  cos  x  =  1 
can  be  reduced  to  the  form 

(cos  x  +  l)(cos  x  —  |)  =  0 


V,  §89] 


TRIGONOMETRY 


109 


, 
\ 


0-  \ 


f,l3 


by  replacing  sin2  x  by  1  —  cos2  x,  transposing  all  the  terms  to 
the  left  side,  and  factoring. 

8Q.  Graphs  of  the  Trigonometric  Functions.  The  varia- 
tion in  the  sine  of  a  given  angle  as  the  angle  increases  from 
0°  to  360°  may  be  exhibited  graphically  as  follows. 

Divide   the   circumference   of   a   unit  „  1X    & 

circle  into  a  convenient  number  of  equal 
arcs.  In  Fig.  43,  the  points  of  division 
are  marked  0,  1,  2,3,  •••  12.  The 
length  of  the  circumference  is  approxi- 
mately 6.3;  lay  this  off  on  the  z-axis 
(Fig.  44)  and  divide  it  into  the  same 
number  of  equal  parts  and  number  them 
to  correspond  with  the  points  of  division  on  the  circumference. 

At  each  point  of  division  on  the  a>axis  lay  off  vertically  the 
line  representation  QP,  of  the  sine  of  the  angle  whose  terminal 
side  goes  through  the  corresponding  point  of  division  on  the 
circle.  Connect  the  ends  of  these  perpendiculars  by  a  smooth 
curve.  This  is  called  the  sine  curve  or  the  graph  of  sin  x. 


FIG.  43 


* 


A 


23 4 5      6\ 


g\  io\   n\   Si2  is    u    is    16    x 


FIG.  44 

As  the  angle  increases  from  0°  to  360°,  P  moves  along  the 
circle  successively  through  the  points  0,  1,  2,  3,  •••,  12,  Q' 
moves  along  the  z-axis  successively  through  the  corresponding 
points  0,  1,  2,  3,  •  •  •,  12,  and  P'  traces  the  sine  curve. 

The  graphs  of  the  other  trigonometric  functions,  cos  x,  tan  x, 


110 


MATHEMATICS 


[V,  §89 


etc.,  are  constructed  in  a  similar  manner  by  making  use  of  their 
line  representations  given  in  §  86. 

If  the  angle  increases  beyond  360°,  P  makes  a  second  revolu- 
tion around  the  circle,  and  the  values  of  all  the  trigonometric 
functions  repeat  themselves  in  the  same  order  and  the  graphs 
from  x  =  6.3  to  x  =  12.6  will  in  all  cases  be  a  repetition  of  those 
from  x  =  0  to  x  =  6.3.  If  P  goes  on  indefinitely  the  graph 
will  be  repeated  as  many  times  as  P  makes  revolutions. 

Functions  which  repeat  themselves  as  the  variable  or  argu- 
ment increases  are  called  periodic  functions.  The  period  is  the 
smallest  amount  of  increase  in  the  variable  which  produces  the 
repetition  of  the  value  of  the  function.  Thus,  sin  a;  is  a  peri- 
odic function  with  a  period  of  360°,  while  the  period  of  tan  x 
is  180°. 

90.  Functions  of  Negative  Angles.  Let  AOC  =  </>  be  any 
angle  placed  on  the  axes;  and  let  AOC'  be  its  negative,  —  <j>; 


FIG.  45 

lay  off  OP'  =  OP  and  draw  PPf.  Let  x,  y  be  the  coordinates 
of  P  and  x',  y'  those  of  P';  let  OP  =  r  and  OP'  =  r'.  Then 
no  matter  what  the  magnitude  or  sign  of  4>, 


y  =  -  y , 


r  =  r' 


V,   §91] 


TRIGONOMETRY 


111 


and  by  the  definitions,  §  83 


sin  (  —  0)  =  —  =  --  =  —  sin 


> 

cos  (—  $)  =-7  =  -  =  cos  </», 
r       r 


tan  (-  <j>)  =    7  =  -  -  =  -  tan 


ctn  (—$)=  —  =  --  =  —  ctn  </>, 

y         y 


sec  (—  <f>)  =—,=-  =  sec 
a;      x 


r  r 

esc  (  —  0)  =  —  =  --  =  —  esc  </>. 


91.  The  Trigonometric  Functions  of  90°  -f  <t>.     Let  any 

angle  </>  be  placed  on  the  axes;  draw  a  circle,  center  at  the  origin, 
with  any  convenient  radius  r,  cutting  the  terminal  side  of  0  in  P 
and  the  terminal  side  of  <f>  +  90°  in  Q.  Let  the  coordinates  of  P 
be  (a,  &);  then  no  matter  in  what  quadrant  P  is,  Q  is  in  the 
next  quadrant  and  its  coordinates  are  (—6,  a),  for  the  right 
triangles  OMP  and  QNO  have  the  hypotenuse  and  an  acute 
angle  of  the  one  equal  to  the  hypotenuse  and  an  acute  angle  of 
the  other.  Then  by  the  definitions,  §  83 


C-6,a)\0 


FIG.  46 


MATHEMATICS 


[V,  §91 


sin  (90°  +  0)  =  -  =  cos  0 
r 


cos  (90°  +  0)  = 


-  b 


=  —  sin  0, 


tan  (90°  +  0)  = 


ctn  (90°  +  0)  = 


-  6 

-  6 


=   —  ctn  0, 


=  —  tan  0, 


sec  (90°  +  0)  = 


=    —  CSC  0, 


csc  (90°  +  0)  =  -  =  sec  0. 
a 

These  formulas  hold  for  all  angles.* 

92.  Functions  of  ±  9,  90°  ±  6,  180°  ±  0,  270°  ±  6.  If  we 
put  for  0  in  succession,  -  6,  6,  90°  -  6,  90°  +  6,  180°  -  0, 
180°  +  6,  270°  -  9,  270°  +  6,  we  obtain  the  values  in  the 
following  table,  6  being  any  angle.*  By  drawing  diagrams  the 
results  tabulated  can  be  verified.  The  student  is  advised  to 
do  this. 


90°—  e. 

90°+0. 

180°—  e. 

180°  +0. 

270"—  8. 

270°  +8. 

360°—  9. 

-e. 

sin 

cos  0 

cos  9 

sin  0 

—  sin  0 

—  cos  6 

—  cos  0 

—  sin  0 

—sin  0 

cos 

sin  0 

—  sin  6 

—  cos  0 

—  cos  6 

—  sin  0 

sin  0 

cos  0 

cos  0 

tan 

ctn  6 

—  ctn  0 

—  tan0 

tan  0 

ctn  e 

—  ctn  0 

—  tan  0 

—  tan  0 

ctn 

tan  0 

—  tan  0 

—  ctn  6 

ctn  6 

tan  0 

—  tan0 

—  ctn  0 

—  ctn  0 

sec 

csc  0 

—  csc  0 

—  sec  0 

—  sec  0 

—  csc  0 

csc  0 

sec  0 

sec  0 

csc 

sec  B 

sec  0 

csc  0 

—  csc  6 

—  sec  5 

—  sec  0 

—  csc  0 

—  csc  0 

If  we  inspect  the  table  carefully,  we  find  that  it  can  be  summed 
up  in  the  two  rules  that  follow. 


*  Except  that  no  angle  whose  terminal  side  falls  on  the  y-axis  has  a  tangent  or 
secant  and  no  angle  whose  terminal  side  falls  on  the  z-axis  haa  a  cotangent  or  cosecant. 


V,  §93] 


TRIGONOMETRY 


113 


1.  Determine  the  sign  by  the  quadrant  in  which  the  angle  would 
lie  if  8  were  acute;  the  result  holds  whether  6  is  acute  or  not. 

2.  //  90°  or  270°  is  involved,  the  function  changes  name  to  the 
corresponding  cof unction,  while  if  180°  or  360°  is  involved  the 
function  does  not  change  name. 

EXAMPLE  1.  sin  177°  =  sin  (180°  -  3°)  =  +  (rule  1)  sin  (rule  2)  3°. 
EXAMPLE  2.  cos  177°  =  cos  (90°  +  87°)  =  -  (rule  1)  sin  (rule  2)  37°. 
EXAMPLE  3.  tan300°  =  tan  (180°  + 120°)  =  +  (rule  1) tan  (rule2)  120°. 

93.  Plotting  Graphs  from  Tables.  For  many  purposes, 
such  as  the  measurement  of  arcs  and  the  speed  of  rotations,  and 
generally  in  the  calculus  and  higher  mathematics,  angles  are 
measured  in  terms  of  a  unit  called  the  radian. 

A  radian  is  a  positive  angle  such  that  when  its  vertex  is  placed 
at  the  center  of  a  circle  the  intercepted  arc  is  equal  in  length 
to  the  radius.  This  unit  is  thus  a  little  less  than  one  of  the 
angles  of  an  equilateral  triangle,  57°. 3  approximately.  It  is 
easy  to  change  from  radians  to  degrees  and  vice  versa,  by 
remembering  that 
(22)  TT  radians  =  180  degrees. 

Unless  some  other  unit  is  expressly  stated,  it  is  always  under- 
stood that  in  graphs  of  the  trigonometric  functions  the  radian 
is  the  unit  angle  and  that  1  unit  on  the  x-axis  represents  1  radian. 
These  graphs  can  be  constructed  from  a  table  of  their  values 
such  as  Table  III  at  the  end  of  the  book.  Thus  to  plot  the 
graph  of  sin  x,  draw  a  pair  of  rectangular  axes  on  squared  paper 


FIG.  47 


114  MATHEMATICS  [V,  §93 

and  mark  the  points  1,  2,  3,  •  •  •  on  the  x-axis.  These  unit 
lengths  are  divided  by  the  rulings  of  the  cross-section  paper 
into  tenths.  At  each  of  these  points  of  division  on  the  x-axis 
lay  off  parallel  to  the  y-axis  the  sine  of  the  angle  from  the  table, 
e.  g.,  at  1  we  plot  AP  =  .84  =  sin  1  (radian).  The  curve  may 
be  extended  beyond  the  first  quadrant  by  the  principles  of  §  92. 
Similarly  the  graphs  of  cos  x  and  tan  x  can  be  plotted  from 
their  tabulated  values. 

EXERCISES 

1.  Express  each  of  the  following  functions  as  functions  of  angles 
less  than  90°. 

(a)  sin  172°,     (6)  cos  100°,     (c)  tan  125°,     (d)  ctn  91°,       (e)  sec  110°, 
(/)  esc  260°,     (g)  sin  204°,     (h)  cos  359°,     (i)  tan  300°,     (j)  ctn  620°. 

2.  Express  each  of  the  preceding  functions  as  functions  of  an  angle 
less  than  45°. 

3.  Express  each  of  the  following  functions  in  terms  of  the  functions 
of  positive  angles  less  than  45°. 

(a)   sin  (-160°),  (6)  cos  (- 30°),  (c)  esc  92°  25', 

(d)  sec  299°  45',  (e)   sin  (-  52°  37'),          (/)  cos  (-  196°  54'), 

(g)    tan  269°  15',  (h)  ctn  139°  17',  (i)  sec  (-  140°), 

(j)   ctn  (-  240°),  (ft)  esc  (-  100°),  (Z)  sin  (-  300°), 

(m)  cos  117°  17',  (n)  sin  143°  21'  16",         (o)  tan  317°  29'  31", 

(p)  ctn  90°  46'  12",  (q)  sec  (-  135°  14'  11"),  (r)  cos  (-  428°). 

4.  Simplify  each  of  the  following  expressions. 

(a)  sin  (90°  +  x)  sin  (180°  +  x)  +  cos  (90°  +  x)  cos  (180°  -  x). 
(6)  cos  (180°  +  x)  cos  (270°  -  y)  -  sin  (180°  +  x)  sin  (270°  -  y). 
(c)  sin  420°  cos  390°  +  cos  (-  300°)  sin  (-  330°). 

5.  Prove  each  of  the  following  relations, 
(a)  cos  \(x  -  270°)  =  +  sin  x/3. 

(6)  sec  ( —  x  —  540°)  =  —  sec  x. 

6.  Verify  each  of  the  following  equations, 
(a)   cos  570°  sin  510°  -  sin  330°  cos  390°  =  0. 

(6)   cos  (90°  +  a)  cos  (270°  -  a)  -  sin  (180°  -  a)  sin  (360°  -  a) 

=  2  sin2  a. 


V,  §93]  TRIGONOMETRY  115 

(c)  3  tan  210°  +  2  tan  120°  =  -  >/3. 

(d)  5  sec2  135°  -  6  ctn2  300°  =  8. 

(e)  sin  (90°  +  «)  sin  (180°  +  x)  +  cos  (90°  +  x)  cos  (180°  -  x)  =  0. 

tan  (90°  +  tt)  +  C8c2  (270°  -«)  =  L+  ^c2  «. 


7.  Construct  a  table  containing  the  functions  of  the  eighths  and 
twelfths  of  360°. 

8.  In  each  of  the  following  equations  find  graphically  the  two  solu- 
tions which  are  between  0°  and  360°  and  compute  the  values  of  the 
other  five  functions  of  each  of  these  angles. 

(a)  sin  x  =  3/5.  (6)  sin  x  =  —  1/3.  (c)   cos  x  =  —  1/3. 

(d)  ctnx  =  -  3.  (e)   sec  x  =  -  5/3.  (/)  esc  x  =  13/5. 

(0)  esc  x  =  —  -^3.          (h)  tan  x  =  —  V?.  (i)    tan  x  =  2.5. 

9.  Verify  each  of  the  following  equations. 

(a)  sin  90°  +  cos  180°  =  0.  (g)  sec  270°  +  esc  0°  =  0. 

(6)  sin  270°  +  cos  0°  =  0.  (h)  sin  120°  +  sin  300°  =  0. 

(c)  esc  90°  +  sec  180°  =  0.  (i)   cos  150°  +  cos  330°  =  0. 

(d)  esc  270°  +  sec  0°  =  0.  (j)   tan  135°  +  tan  225°  =  0. 

(e)  sin  0°  +  cos  270°  =  0.  (fc)  ctn  315°  +  ctn  45°  =  0. 
(/)  sin  180°  +  cos  90°  =  0.  (1)    sin  120°  +  cos  210°  =  0. 

10.  Find  graphically  another  angle  between  0°  and  360°  which  has 
the  same 

(a)  sine  as  140°,  (6)  sine  as  220°,  (c)  cosine  as  330°, 

(d)  tangent  as  230°,         (e)  cotangent  as  110°,          (/)  secant  as  160°. 

11.  Find  the  values  of  6  between  0°  and  360°  which  satisfy  the 
following  equations. 

(a)  sin  0  =  sin  320°.  (d)  cos  0  =  -  cos  50°. 

(6)   tan  9  =  tan  125°.  (c)   ctn  0  =  -  ctn  220°. 

(c)  sec  0  =  sec  80°.  (/)   esc  0  =  -  esc  340°. 

12.  In  what  quadrant  does  an  angle  lie  if  sine  and  cosine  are  both 
negative?     if  cosine  and  tangent  are  both  negative?     if  cotangent  is 
positive  and  sine  negative? 

13.  In  finding  cos  x  from  the  equation  cos  x  =  =*=  Vl  —  sin2  x, 
when  must  we  choose  the  positive  and  when  the  negative  sign  ? 

14.  Plot  the  graphs  of  each  of  the  following  functions  and  determine 
its  period. 


116 


MATHEMATICS 


[V,  §  93 


(a)  cos  x. 
(d)  sec  x. 
(g)  cos  (-  x). 


(6)  tan  x. 
(e)  esc  x. 
(K)  sin  (90°  +  x). 


(c)  ctn  x. 
(/)   sin  (-  x). 
(i)  sin  x  —  cos  x. 


15.    Plot  the  graph  of  each  of  the  following  functions. 

(a)  x  +  sin  x.  (6)  x2  +  sin  x.  (c)  sin  x  +  cos  x. 

(d)  x  +  cos  x.  (e)  x  —  cos  x.  (/)  x  —  1  +  sin  x. 

94.   Sine  and  Cosine  of  the  Sum  of  two  Angles.     Let 

AOB  =  x,  BOC  =  y,  then  AOC  =  x  +  y.  With  0  as  center 
and  a  convenient  radius  r  >  0,  strike  an  arc  cutting  OC  in  P. 
Drop  PQ  perpendicular  to  OB,  also  PR  and  QS  perpendicular  to 


o      R      S 


it      o      s 


FIG.  48 


CM.     Through  Q  draw  a  parallel  to  OA  cutting  Pfl  in  T.     Then 
by  (7),  §  75, 

r  sin  (x  +  y)  =  RP  =  SQ  +  TP. 
Now  by  (7)  and  (8),  §  75,  we  have 

OQ  =  r  cos  y  and  <SQ  =  OQ  sin  x  =  r  cos  y  sin  x, 
PQ  -  r  sin  i/  and  TP  =  PQ  cos  x  =  r  sin  y  cos  2. 

Hence  we  may  write 

r  sin  (a:  +  y)  =  r  cos  y  sin  x  +  r  sin  y  cos  a:, 
and 
(23)  sin  (x  +  y)  =  sin  x  cos  y  +  cos  x  sin  y. 


V,  §95]  TRIGONOMETRY  117 

Similarly,  we  may  write 

r  cos  (x  +  y)  =  OR  =  OS  -  TQ. 
Then  as  before, 

OS  =  OQ  cos  x  =  r  cos  y  cos  x, 
TQ  =  PQ  sin  x  =  r  sin  y  sin  x. 

Hence  we  may  write 

r  cos  (x  +  T/)  =  r  cos  ?/  cos  x  —  r  sin  y  sin  .r, 
and 

(24)  cos  (x  +  y)  =  cos  x  cos  y  —  sin  x  sin  y. 

The  above  formulas,  therefore,  hold  true  for  all  acute  angles 
x  and  y.  They  are  called  the  addition  formulas. 

It  is  readily  proved  that  if  x  =  a  and  y  =  /3  are  any  two 
acute  angles  for  which  these  formulas  hold  good  they  will  hold 
good  for  any  two  of  the  angles  a,  ft,  a  +  90°,  a  -  90°,  /3  +  90°, 
/3  —  90°.  Therefore,  since  we  have  found  that  they  hold  good 
for  all  acute  angles,  they  hold  good  for  all  positive  or  negative 
angles  of  any  magnitude  whatever. 

The  addition  formulas  may  be  translated  into  words  as  follows: 

I.  The  sine  of  the  sum  of  two  angles  is  equal  to  the  sine  of  the 
first  times  the  cosine  of  the  second,  plus  the  cosine  of  the  first  times 
the  sine  of  the  second. ' 

II.  The  cosine  of  the  sum  of  two  angles  is  equal  to  the  cosine  of 
the  first  times  the  cosine  of  the  second  minus  the  sine  of  the  first 
times  the  sine  of  the  second. 

95.  Tangent  of  the  Sum  of  two  Angles.  This  can  be  de- 
rived from  the  addition  formulas  as  follows 

sin  (x  +  y)       sin  x  cos  y  +  cos  x  sin  y 

tan  (x  +  y)  =  -  — : —  — : —  . 

cos  (x  +  y)       cos  x  cos  y  —  sin  x  sin  y 

If  we  divide  each  term  of  the  numerator  and  denominator  of 


118  MATHEMATICS  [V,  §95 

the  last  fraction  by  cos  x  cos  y,  we  have 

sin  x       sin  y 

cosx       cos  y 
tan  (x  +  y)  = 


sin  x  sm 


cos  z  cos  y 
that  is 

(25)  «»(.  +  »)-    *•"  +  «"'»  . 

1  —  tan  x  tan  y 

This  formula  holds  good  for  all  angles  such  that  z,  y,  and  z  +  y 
have  tangents. 

96.  Functions  of  Twice  an  Angle.     If  we  put  z  for  y  in 
(23),  (24),  §  94,  and  (25),  §  95,  these  formulas  give 

(26)  sin  2z  =  2  sin  z  cos  z. 

(27)  cos  2z  =  cos2  z  —  sin2  z. 

(28)  =  2  cos2  z  -  1. 

(29)  =1-2  sin2  z. 

2  tan  z 


(30)  tan2x  = 


1  —  tan2  x 


97.  Functions  of  Half  an  Angle.  The  preceding  formulas 
are  true  for  all  values  of  x  for  which  they  have  a  meaning.  Hence 
we  may  replace  x  by  any  other  quantity.  If  we  write  x/2  in 
place  of  x  in  (28)  and  (29),  §  96,  and  solve  the  resulting  equa- 
tions for  sin  (z/2)  and  cos  (z/2),  we  find 


,  ._  —  cos  z 

(31)  sm  \x  =  db 


.                 /I  +  cos  z 
(32)  cos  \x  =  db  ^ g ' 

Whence  on  dividing  (31)  by  (32) 

fl  —  cos  z       1  —  cos  z  sin  z 


(33)    tan  |z  =  ±  x/ , 

1    cos  z  sin  z  1  +  cos  z 


V,  §97]  TRIGONOMETRY  119 

The  positive  or 'negative  sign  is  to  be  chosen  according  to  the 
quadrant  in  which  z/2  lies. 

EXERCISES 

1.  Putting  75°  =  45°  +  30°,  find  cos  75°  and  tan  75°. 

2.  ^Putting  15°  =  45°  +  (-  30°),  find  sin  15°,  cos  15°,  and  tan  15°. 

3.  'Putting  15°  =  60°  +  (-  45°),  find  sin  15°,  cos  15°,  and  tan  15°. 

4.  Putting  90°  =  60°  +  30°,  find  sin  90°  and  cos  90°. 

5.  Show  that  sin  (x  —  y)  =  sin  x  cos  y  —  cos  x  sin  y. 

6.  Show  that  cos  (x  —  y)  =  cos  x  cos  y  +  sin  x  sin  y. 

7.  Putting  15°  =  60°  -  45°,  find  sin  15°. 

8.  Show  that  sin  3x  =  sin  x(3  —  4  sin2  x)  =  sin  x(4  cos2  x  —  1). 

9.  Show  that  cos  3x  =  cos  x(4  cos2  x  —  3)  =  cos  z(l  —  4  sin2  x). 

10.  Find  sin  4x;  cos  4x;  tan  4x. 

11.  Show  that  tan  (45°  +  A)  =  ?  +  ^°  AA  . 

1  —  tan  A 

12.  Show  that 

,          .        tan  x  —  tan  y 

(a)  tan  (x  —  y)  =  —  —  , 

1  +  tan  x  tan  y ' 

,          .       ctn  x  ctn  y  —  1 
Ctn(x  +  y)  =  ^nT  +  inT- 

13.  From  the  trigonometric  ratios  of  30°,  find  sin  60°,  cos  60°,  tan  60°. 

14.  Express  sin  6,4,  cos  6 A,  tan  GA  in  terms  of  functions  of  3A. 

15.  Find  sin  22|°,  cos  22-J-0,  and  tan  22£°,  from  cos  45°. 

16.  Find  sin  15°,  cos  15°,  and  tan  15°,  from  cos  30°. 

17.  Find  cos  (x  +  y),  having  given  sin  x  =  3/5  and  sin  y  =  5/13, 
x  being  positive  acute,  y  being  positive  obtuse.  Ans.   —  63/65. 

18.  Verify  the  following: 

(a)  sin  (60°  +  x)  -  sin  (60°  -  x)  =  sin  x. 

(6)  cos  (30°  +  y)  -  cos  (30°  -  y)  =  -  sin  y. 

(c)  cos  (45°  +  x)  +  cos  (45°  —  x)  =  V2  cos  x. 

(d)  cos  (Q  +  45°)  +  sin  (Q  -  45°)  =  0. 

(e)  sin  (x  +  y)  sin  (x  —  y)  =  sin2  x  —  sin2  y. 

.  , ,  sin  (x  +  y)      tan  x  +  tan  y  2  tan  x 

(f )  ~ — 7 •       (0)  sin  2x  = 

^•*    '    otr\     tfm    1*1  -for*    />-    *.,»!,!  ™' 


sin  (x  —  y)      tan  x  —  tan  y '  1  +  tan2  x ' 

,, ,  esc2  x  sin  $x 

(h)   sec  2x  =  — ^.  (i)    tanjx  =  —         =— r-. 

esc2  x  —  2  1  +  cos  \x 

(fi    ctn  ix  -      sin^  (H  t«n  '  A   -  l  ~  C08  A 
n*X       1-cos^x*  sin  A      ' 


120  MATHEMATICS  [V,   §97 

(I)    2  esc  2s  =  sec  s  esc  s. 

(m)  tan  (x  +  45°)  +  ctn  (x  -  45°)  =  0. 

19.  Prove  each  of  the  following  identities, 
(a)   cos  (A  +  B)  cos  (A  -  B)  =  cos2  A  -  sin2  B. 
(6)    sin  (A  +  B)  cos  B  —  cos  (A  +  B)  sin  B  =  sin  A. 

(c)  sin  (A  +  B)  +  cos  (A  -  B)  =  (sin  A  +  cos  A) (sin  B  +  cos  B). 

(d)  cos4  A  =  f  +  5  cos  2A  +  |  cos  4A. 

(e)  sin4  A  =  f  —  5  cos  2A  +  |  cos  4A. 
(/)  sin2  A  cos2  A  =  |  -  |  cos  4A. 

(g)   sin2  .A  cos4  A.  =  ^  +  jj  cos  2A  —  ^  cos  4A  —  ^  c°s  6A. 
(/i)   cos  (x  —  y  -\-  z)  =  cos  a;  cos  y  cos  2  +  cos  x  sin  ?/  sin  z 

—  sin  x  cos  y  sin  z  +  sin  x  sin  y  cos  z. 

(i)    cos  £  sin  (y  —  z)  +  cos  y  sin  (z  —  x)  +  cos  z  sin  (x  —  y)  =  0. 
0')    sin  A  +  sin  5  =  2  sin  f(A  +  5)  cos  f  (A  -  B). 
(fc)   sin  A  -  sin  B  =  2  cos  i(^  +  B}  sin  |(A  -  B). 
(1)    cos  A  +  cosB  =2  cos  \(A  +  B)  cos  f(A  -  B). 
(m)  cos  A  —  cos  5  =  —  2  sin  |(A  +  .B)  sin  %(A  —  B). 
(n)  sin  A  cos  (B  —  C)  —  sin  £  cos  (A.  —  C)  =  sin  (A.  —  JB)  cos  C. 
(o)    cos2  %<f>(l  +  tan  |0)2  =  1  +  sin  0. 
(p)  sin2  |x(ctn  |x  —  I)2  =  1  —  sin  x. 

,  .  2  sec  A  .  x  .         2  sec  A 

(g)    sec2  JA  =  - — : -.  .  (r)  esc2  §A  = s . 

1  +  sec  A  sec  A  —  1 

98.  Solution  of  Oblique  Triangles.  One  of  the  chief  uses 
of  trigonometry  is  to  solve  triangles.  That  is,  having  given 
three  parts  of  a  triangle  (sides  and  angles)  at  least  one  of  which 
must  be  a  side,  to  find  the  others.  In  plane  geometry  it  has 
been  shown  how  to  construct  a  triangle,  having  given 

CASE  I.     Two  angles  and  one  side. 

CASE  II.     Two  sides  and  the  angle  opposite  one  of  them. 

CASE  III.     Two  sides  and  the  included  angle. 

CASE  IV.     Three  sides. 

When  the  required  triangle  has  been  constructed  by  scale  and 
protractor  the  parts  not  given  may  be  found  by  actual  measure- 
ment. The  results  obtained  by  such  graphic  methods  are  not, 
however,  sufficiently  accurate  for  many  practical  purposes. 


V,  §  99]  TRIGONOMETRY  121 

Nevertheless,  they  are  very  useful  as  a  check  upon  the  com- 
puted values  of  the  unknown  parts.  Other  checks  are  fur- 
nished by  the  theorems  of  plane  geometry  that  the  sum  of  the 
angles  of  any  triangle  is  180°,  and  that  if  two  sides  (angles)  are 
unequal  the  greater  side  (angle)  lies  opposite  the  greater  angle 
(side).  The  properties  of  isosceles  triangles  can  also  be  used  in 
certain  special  cases. 

The  direction  solve  a  triangle  tacitly  assumes  that  a  sufficient 
number  of  parts  of  an  actual  triangle  are  given.  A  proposed 
problem  may  violate  this  assumption  and  there  will  be  no 
solution.  Thus,  there  is  no  triangle  whose  sides  are  14,  24, 
and  40 ;  likewise,  there  is  no  triangle  of  which  two  sides  are  9 
and  10  and  the  angle  opposite  the  former  is  64°  10'.  Any  tri- 
angle which  can  be  constructed  can  be  solved. 

Any  oblique  triangle  can  be  divided  into  right  triangles  by  a 
perpendicular  from  a  vertex  upon  the  opposite  side,  and  this 
method  when  applied  to  the  various  cases  leads  to  three  laws, 
called  the  law  of  sines,  the  law  of  cosines,  and  the  law  of  tan- 
gents, by  means  of  which  the  unknown  parts  of  any  oblique 
triangle  can  be  computed.  We  proceed  to  prove  these 
laws. 

99.  Law  of  Sines.  Any  two  sides  of  a  triangle  are  to  each 
other  as  the  sines  of  the  opposite  angles. 

In  any  oblique  triangle  let  a,  b  and  c  be  the  measures  of  the 
lengths  of  the  sides  and  A,  B,  and  C  the  measures  of  the  angles 
opposite.  Drop  the  perpendicular  CD  =  p  from  the  vertex 
of  angle  C  to  the  opposite  side. 

Two  possible  cases  are  shown  in  Figs.  49,  50.  In  either  of 
these  figures, 

p  =  b  sin  A. 

In  Fig.  49, 

p  =  a  sin  B. 


122 


MATHEMATICS 


[V,   §99 


In  Fig.  50, 

p  =  a  sin  (180°  -  B)  =  a  sin  B. 

Therefore,  whether  the  angles  are  all  acute,  or  one  is  obtuse 
a  sin  B  =  b  sin  A, 


D     B 


FIG.  49 


whence  dividing  first  by  sin  A  sin  B,  and  second  by  6  sin  B, 


(34) 


sin  A       sin  B ' 


or 


sin  A 
sin  B ' 


Similarly,  by  drawing  perpendiculars  from  A  and  B  to  the 
opposite  sides,  we  obtain 

be  a  c 


Hence, 

(35) 


sin  B      sin  C"         sin  A       sin  C ' 

a  b  c 

sin  A       sin  B       sin  C  * 


It  is  evident  that  a  triangle  may  be  solved  by  the  aid  of  the 
law  of  sines  if  two  of  the  three  known  parts  are  a  side  and  its 
opposite  angle.  The  case  of  two  angles  and  the  included  side 
being  given,  may  also  be  brought  under  this  head,  since  we 
may  find  the  third  angle  which  lies  opposite  the  given  side. 

100.  Law  of  Cosines.  In  any  triangle,  the  square  of  any 
side  is  equal  to  the  sum  of  the  squares  of  the  other  two  sides  minus 
twice  the  product  of  these  two  sides  into  the  cosine  of  their  included 
angle. 


V,  §  100] 


TRIGONOMETRY 


123 


Let  ABC  be  any  triangle.     Drop  a  perpendicular  BD  from  B 
on  AC  or  AC  produced.     Two  possible  cases  are  shown  in 


FIG.  51 
Figs.  51,  52.     Then  we  have  either 


or  else 


and 


CD  =  b  -  AD  (Fig.  51), 
=  6  —  c  cos  A, 

CD  =  b  +  AD  (Fig.  52) 
=  6  +  c  cos  (180°  -  A) 
=  6  —  c  cos  A, 

p  =  c  sin  A  (Fig.  51), 

p  =  c  sin  (180°  -  A)  =  c  sin  A  (Fig.  52). 


Hence,  in  either  figure,  we  may  write 

CD  =  b  —  c  cos  A         and         p  =  c  sin  A. 
Again,  in  either  figure, 
a2  =  CD2  +  P2 

=  (b  —  c  cos  A)2  -f-  (c  sin  A)- 

=  b2  -  2bc  cos  A  +  c2  (sin2  A  +  cos2  A) 

=  b-  —  2bc  cos  A  +  c2 


that  is 
(36) 


COS 


In  like  manner  it  may  be  proved  that  the  law  of  cosines  applies 
to  the  side  b  or  to  the  side  c. 


124  MATHEMATICS  [V,  §  100 

These  formulas  may  be  used  to  find  the  angles  of  a  triangle 
when  the  three  sides  are  given  and  also  to  find  the  third  side 
when  two  sides  and  the  included  angle  are  given. 

101.  Law  of  Tangents.  The  sum  of  any  two  sides  of  a  tri- 
angle is  to  their  difference  as  the  tangent  of  half  the  sum  of  their 
opposite  angles  is  to  the  tangent  of  half  their  differ&nce. 

From  the  law  of  sines,  we  have 

a       sin  A 
b  =  sin  B' 

whence,  by  division  and  composition  in  proportion,  we  find 

o  +  b  _  sin  A  +  sin  B 
a  —  b       sin  A  —  sin  B  ' 

Let  x  +  y  =  A  and  x  —  y  =  B.     Then  we  have 

2x  =  A  +  B,        and        x  =  f  (A  +  B), 
2y  =  A  -  B,         and        y  =  %(A  -  B). 

Hence,  substituting  in  (37),  we  find 

sin  A  +  sin  B       sin  (x  +  y)  +  sin  (x  —  y) 
sin  A  —  sin  B       sin  (x  +  y)  —  sin  (x  —  y) 

_  2  sin  x  cos  y  _  tan  x 
2  cos  x  sin  y       tan  y 

_  tan  |(A  +  B) 

~  tan  |(A  -  B} ' 

From  (37)  and  the  preceding  result,  we  have 

(38)  °  +  b  =  tan  %(A  +  B) 
a-  b      tan  %(A  -  B) ' 

Since 

tan  \(A  +  B)  =  tan  |(180°  -  (7)  =  tan  (90°  -  £C)  =  ctn  |C, 
we  may  write  the  law  of  tangents  in  the  form 

(39)  tan  %(A  -  B)  =  ?—^-ctn  \C. 

a  +  o 


V,  §103]  TRIGONOMETRY  125 

As  a  check,  (38)  is  the  more  convenient  form,  while  for  solving 
triangles,  (39)  is  preferred  by  some  computers.  If  6  >  a,  then 
B  >  A.  The  formula  is  still  true,  but  to  avoid  negative  num- 
bers the  formula  in  this  case  should  be  written  in  the  form 

.  b  +  a  =  tan  %(B  +  A) 

b  -  a  ~  tan  \(B  -  A) ' 

When  two  sides  and  the  included  angle  are  given,  as  a,  6,  C, 
the  law  of  tangents  may  be  employed  in  finding  the  two  unknown 
angles  A  and  B. 

102.  Methods  of  Computation.     The  method  to  be  used  in 
computing  the  unknown  parts  of  a  triangle  depends  on  what 
parts  are  given.     In  what  follows  triangles  are  classified  ac- 
cording to  the  given  parts  and  the  methods  of  computation  are 
stated  and  illustrated  by  examples. 

103.  Case  I.     Given  two  Angles  and  one  Side.     There  is 
always  one  and  only  one  solution,  provided  the  sum  of  the 
given  angles  is  less  than  180°. 

The  third  angle  is  found  by  subtracting  the  sum  of  the  two 
given  angles  from  180°.  The  unknown  sides  are  found,  suc- 
cessively, by  the  law  of  sines. 

EXAMPLE.  In  a  triangle  given  two  angles  38° 
and  75°  43',  and  the  side  opposite  the  former 
180;  find  the  other  parts. 

Construct  the  triangle  approximately  to  scale 
and  denote  the  unknown  parts  by  suitable  let- 
ters as  in  Fig.  53.  £  A 

First  compute  the  third  angle  C  =  66°  17'.  FIG.  53 

To  compute  b  use  the  law  of  sines, 

_6_      sin  75°  43' 
180  ™     sin  38°     * 

In  any  proportion  imagine  the  means  and  the  extremes  to  be  paired 
by  lines  crossing  at  the  equal  sign, 


126  MATHEMATICS  [V,  §103 


then  the  rule:  Multiply  the  pair  of  knowns  and  divide  by  the  known  in 
the  other  pair;  or,  Add  the  logarithms  of  the  pair  of  knowns  and  the  co- 
logarithm  of  the  known  in  the  other  pair. 

FIRST  METHOD:  without  logarithms. 

sin  75°  43'  =      0.9691 
180 


775280 
9691 

sin  38°  =  0.6157)174.4380(283.3 
12314 


51298,  etc. 
whence  6  =  283.3. 

SECOND  METHOD:  with  logarithms. 

log  180  =  2.2553 

log  sin  75°  43'  =  9.9864  -  10 

colog  sin  38°  =  0.2107 

log  b  =  2.4524 

18 


15)60(4  b  =  283.4. 

Similarly  we  may  compute  c.  Using  logarithms,  we  find  c  =  267.7. 
Not  using  logarithms,  we  find  267.6.  The  difference  in  the  two  answers 
is  due  to  the  slight  inaccuracy  caused  by  our  using  only  four  decimal 
places. 

EXERCISES 

1.  Given  two  angles  43°  and  67°  and  the  included  side  51;  find  the 
other  parts.  Ans.  70°,  49.96,  37.02. 

2.  Given  two  angles  24°  14'  and  43°  13'  and  the  side  opposite  the 
latter  240;  find  the  other  parts.  Ans.  112°  33',  143.9,  323.8. 

3.  Solve  the  triangle  ABC  being  given  A  =  17°  17',  B  =  102°  25', 
and  a  =  36.84.  Ans.  C  =  60°  18',  c  =  107.7,  6  =  121.1. 

4.  Solve  the  triangle  LMN  being  given  L  =  28°,  M  =  51°,  I  =  6.3. 

Ans.  N  =  101°,  n  =  13.17,  m  =  10.43. 


V,  §104] 


TRIGONOMETRY 


127 


104.  Case  II.  Given  two  Sides  and  the  Angle  opposite 
one  of  Them.  This  case  sometimes  admits  two  solutions  and 
on  this  account  is  called  the  ambiguous  case.  The  number  of 
solutions  can  be  determined  by  constructing  the  triangle  to 
scale  as  follows. 

To  fix  our  ideas,  let  the  given  angle  be  A,  the  given  opposite 
side  a,  and  the  given  adjacent  side  6.  Construct  the  given 
angle  A,  and  on  one  of  its  sides  lay  off  AC  =  b,  the  given 
adjacent  side,  and  drop  a  perpendicular  CP,  of  length  p,  from 
C  to  the  other  side  of  the  given  angle  A.  With  C  as  center 
and  with  radius  o,  the  given  opposite  side,  strike  an  arc  to 
determine  the  vertex  of  the  third  angle  B.  Several  possible 
cases  are  shown  in  Fig.  54. 


t.  One  Solution 


S,  One  .Solution 


FIG.  54 


A  study  of  these  diagrams  shows  that  there  will  be  two 
solutions  when,  and  only  when,  the  given  angle  is  acute  and  the 
length  of  the  given  opposite  side  is  intermediate  between  the 
lengths  of  the  perpendicular  and  the  given  adjacent  side;  that  is 

A  <  90°         and         p  <  a  <  b. 

The  two  triangles  to  be  solved  are  AB\C  and  AB2C.     Since 


128 


MATHEMATICS 


[V,  §104 


the  triangle  BiCB2  is  isosceles,  the  obtuse  angle  BI  (i.  e.,  angle 
ABiC)  is  the  supplement  of  the  acute  angle  B-2. 

The  following  examples  illustrate  the  method  of  computing 

the  unknown  parts  in  Case  II. 

EXAMPLE  1.  One  angle 
of  a  triangle  is  34°  23',  the 
side  opposite  is  44.24  and 
another  side  is  60.35;  find 
the  other  parts. 

On   constructing   the   tri- 

j.  B^~  p  <B angle  to  scale  as  in   Fig.  55, 

pIG    55  it  appears  that  there  are  two 

solutions.     This  is  verified  by 

computing  p  =  60.35  sin  34°  23'.     Noting  from  the  tables  that  sin  35°  <  .6, 
it  is  evident  that  p  <  40. 

Let  us  solve  first  the  triangle  AB2C,  the  angle  B2  being  acute.     By 
the  law  of  sines, 


60.35          sin 


44.24      sin  34°  23' 
B2  =  50°  23' 


log  60.35  =  1.7807 
s  sin  34°  23'  =  9.7518  -  10 
colog  44.24  =  8.3542  -  10 
log  sin  52  =  9.8867  -  10 
64 


10)30(3 


Then  find  C2  (i.  e.,  angle  ACBJ  =  95°  14'.     To  find  c2  (i.  e.,  side 
use  the  law  of  sines  again, 

c2          sin  95°  14' 


44.24       sin  34°  23' 


c2  =78.02 


log  44.24  =  1.6458 
log  sin  95°  14'  =  9.9982  -  10 
colog  sin  34°  23'  =  0.2482 
log  c2  =  1.8922 
21 


6)10(2 


To  solve  the  triangle  AB&,  we  first  find  BI  =  129°  37'  being  the 
supplement  of  52,  and  then  the  third  angle  Ci  =  16°  00'.  To  find  Ci 
(i.  e.,  the  side  ABi)  use  the  law  of  sines, 


V,   §104] 


TRIGONOMETRY 


129 


C] 


sin  16° 


44.24       ein  34°  23' 


d  =  21.59 

CHECK. 

c2  =  78.02 
ci  =  21.59 


log  44.24  =  1.6458 
log  sin  16°  =  9.4403  -  10 
colog  sin  34°  23'  =  0.2482 
log  ci  =  1.3343 


=  2(44.24  cos  50°  23') 


c2  - 


=  56.43 


log  2  =  0.3010 
log  44.24  =  1.6458 
log  cos  50°  23'  =  9.8046  -  10 
log  £i£2  =  1.7514 


=  56.41 


EXAMPLE  2.     One  angle  of  a  triangle  is  34°  23',  the  side  opposite  is 
60.35  and  another  side  is  44.24.     Solve. 

There  is  only  one  solution 
as  shown  by  constructing. 

44.24  sing 

60.35  ~  sin  34°  23' ' 

whence  B  =  24°  27'  and  the 
third  angle  C  =  121°  10'. 

c 


FIG.  56 


sin  121°  10' 


60.35       sin  34°  23'  ' 
whence  c  =  91.46 

EXERCISES 

1.  Two  sides  of  a  triangle  are  17.16  and  14.15  and  the  angle  opposite 
the  latter  is  42°.     Find  the  other  parts. 

Ans.  125°  46',  12°  14',  4.483,  or  54°  14',  83°  46',  21.02 

2.  In  the  triangle  AGK,  A  =  31°  14',  a  =  54,  g  =  48.6.     Find  the 
other  parts.  Ans.  27°  49',  120°  57',  89.3 

3.  A  50  ft.  chord  of  a  circle  subtends  an  angle  of  100°  at  the  center. 
A  triangle  is  to  be  inscribed  in  the  larger  segment  having  one  side 
40  ft.  long.     How  long  is  the  third  side?     How  many  solutions? 

Ans.  65.22 

4.  If  the  triangle  of  Ex.  3  is  to  have  one  side  60  ft.  long,  how  many 
solutions?     How  long  is  the  third  side.  Ans.  18.88  or  58.25 

10 


130 


MATHEMATICS 


[V,  §104 


105.  Case  III.     Given  two  Sides  and  the  included  Angle. 

There  is  always  one  and  only  one  solution.  The  third  side 
can  be  found  by  the  law  of  cosines  and  if  the  angles  are  not 
required,  this  is  a  convenient  method  of  solution,  especially  if 
the  given  sides  are  not  large. 

EXAMPLE  1.     Two  sides  of  a  triangle  are  2.1  and  3.5  and  the  in- 
cluded angle  is  53°  8'.     Find  the  third  side. 

x2  =  27P  +  iTB2  -  2  (2.1)  (3.5)  cos  53°  8' 

=  4.41  +  12.25  -  14.7  X  0.6000  =  7.84, 
whence  x  =  2.8. 

If  the  other  two  angles  as  well  as  the  third  side  are  required, 
the  two  angles  should  be  found  by  the 
law  of  tangents  and  then  the  third  side 
can  be  found  by  the  law  of  sines. 
Both  these  computations  can  be  made 
by  logarithms. 


EXAMPLE  2.     In  the  triangle  ARK, 
a  =  23.45,  r  =  18.44,  and  K  =  81°  50'. 


Find  the  other  parts. 
By  the  law  of  tangents, 


a+r      tan 


+  R) 


_ 
a  —  r      tan  5  (A  —  R)  ' 

The  actual  computation  may  be  arranged  as  follows. 


a  =  23.45 

r  =  18.44 

a  +  r  =  41.89 

a  -  r  =    5.01 


180°  00' 

K  =    81°  50' 

A  +  R  =    98°  10' 

1(A  +  R)  =    49°    5' 


41.89 


tan  49°  5' 


5.01        tan  \(A-R) 

log  5.01  =  0.6998 
log  tan  49°  5'  =  0.0621 

colog  41.89  =  8.3779  -  10 
log  tan  $(A  -  R)  =  9.1398  -  10 

i(A  -  R)  =    7°  51' 
HA  +  B)  =  49°    5' 


A  =  56°  56' 


R  =  41°  14' 


V,  §106]  TRIGONOMETRY  131 

CHECK. 

23.45  =  sin  56°  56' 
18.44      sin  41°  14' 

log  23.45  =  1.3701  log  18.44  =  1.2658 

log  sin  41°  14'  =  9.8190  -  10        log  sin  56°  56'  =  9.9233  -  10 
1.1891  1.1891 

To  compute  k  use  the  law  of  sines, 

k          sin  81°  50' 
23.45  ~  sin  56°  56' ' 
whence  k  =27.70 

EXERCISES 

1.  In  the  triangle  ABC  given  a  =  52.8,  b  =  25.2,  C  =  124°  34'; 
find  the  other  parts.  Ans.  38°  15',  17°  11',  70.2 

2.  Given  I  =  131,  m  =  72,  N  =  39°  46',  find  n,  L,  M. 

Ans.  88.57,  108°  54',  31°  20'. 

3.  Given  u  =  604,  v  =  291,  W  =  106°  19',  find  U,  V,  w. 

Ans.  51°  32',  22°  9',  740.4 

4.  To  find  the  distance  between  two  objects  A  and  B,  separated  by 
a  swamp,  a  station  C  is  selected  so  that  CA  =  300  ft.,  CB  =  277  ft., 
and  angle  ACB  =  65°  47',  can  be  measured.     Compute  AB. 

Ans.  313.9 

5.  Two  sides  of  a  parallelogram  are  23.47  and  62.38  and  one  angle 
is  71°  30'.     Find  its  diagonals.  Ans.  59.27  and  73.29 

106.  Case  IV.     Given  the  three  Sides.     There  is  one  and 
only  one  solution,  provided  no  side  is  grea- 
ter than  the  sum  of  the  other  two. 

The  angles  can  be  computed,  in  succes- 
sion, by  the  law  of  cosines. 

EXAMPLE  1.     The  sides  of  a  triangle  are  5,  7, 
8.     Find  the  angles. 

49  =  25  +  64  -  2  X  5  X  8  cos  A, 


132  MATHEMATICS  [V,  §106 

whence  cos  A  =  |,  A  =  60°. 

25  =  49  +  64  -  2  X  7  X  8  cos  B, 
cos  B  =  -B  =  0.7857,        B  =  38°  13'. 
64  =  25  +  49  -  2  X  5  X  7  cos  C, 
cos  C  =  |  =  0.1429,         C  =  81°  47'. 
CHECK.  60°  +  38°  13'  +  81°  47'  =  180°  00'. 

The  law  of  cosines  is  not  adapted  to  logarithms  but  can  be 
transformed  as  follows.  The  three  sides  of  a  triangle  ABC, 
being  given,  then 

a2  =  &2  +  c2  -  2bc  cos  A, 
whence 

62  +  c2  -  a2 
(41)  cosA=--2bT~' 

To  adapt  this  to  logarithmic  computation,  subtract  each 
member  from  unity 

&2  +  c2  -  a2       2bc  -  b2  -  c2  +  a2 


1  —  cos  A  =  1  — 


2bc 

a2  -  (b  -  c)2 


2bc 
Hence  we  have 

(42)       2sin4A  =  l-cosvl=(a  +  fe- 


If  we  now  set  a  +  &  +  c  =  2s,  we  have 

a  +  &  —  c  =  2(s  —  c), 
a  -  6  +  c  =  2(«  -  6). 

Substituting  these  values  in  (42)  we  find 


(43)  sm 

Similarly, 


-  a)(s  -  c)  -rfir      (s  -  a)(g  -  6) 

,         sin'  ^o  =  —  -  -  -.  — 
ac  ab 


V,  §106]  TRIGONOMETRY  133 

Again,  adding  each  member  of  (41)  to  unity, 

52  +  C2  _  Q2       (6  +  c)2  -  a2 
1  +  cos  A  =  1  +  -   -^-  -2£-  - 

_  (b  +  c  +  a)(&  +  c  -  a) 

26c 
Therefore, 

2  cos*  |A  =  1  +  cos  A  =  2'(  V"  a)  , 

oc 

whence 

(44)  cos*  4A  =  '-^^  . 

Similarly, 

,  ID       «(*  -  6)  ,  ir<      s(s  -  c) 

cos2  §5  —  -          —  ,         cos2  \C  =  --  -  —  . 
ac  ab 

Dividing  sin2  %A  by  cos2  \A,  we  have,  by  (43)  and  (44). 

tan2  %A  =  (s  —  b)(s  -  c)/s(s  -  a) 

=  (s  -  a)(s  -  b)(s  -  c)/s(s  -  a)2. 
It  follows  that 

1         KS  -a)(s  -b)(s-c) 

(45)  tan  \A  =  -     —  \/—  —  . 

s  —  a  \  s 

If  we  now  set 

(46)  r  =  V(*  -  a)(s  -  b)(s  -  c)/s, 

the  equation  (45)  becomes 

(47)  tan  \A  =  —  ^—  . 

s  —  a 

Similarly, 

7"  * 

tan  %B  =  -  tan 


s  —  b  s  —  c 


It  will  be  shown  in  §  107  that  r  is  the  radius  of  the  circle  in- 
scribed in  the  given  triangle. 


134 


MATHEMATICS 


[V,  §  106 


EXAMPLE.     The  sides  of  a  triangle  are  77,   123,   130.     Find  the 
angles. 

Us  -  a)(s  -  b)(s  -c)    log  (s  -  a)  =  1.9445 
=   V  s  log  (s  -  6)  =  1.6232 

log  (s  -  c)  =  1.5441 

colog  s  =  7.7825  -  10 


tan  \A  = 


s  —  a 
a  =  77 
b  =  123 
c  =  130 


2)2.8943 


2s  =  330 

s  =  165 

s  -  a  =    88 

s  -  b  =    42 

s  -  c  =    35 

CHECK  165 


logr  =  1.4472 
log  tan  \A  =  9.5027  -  10 
log  tan  \E  =  9.8240  -  10 
log  tan  \C  =  9.9031  -  10 
\A  =  17°  39' 
\B  =  33°  42' 
\C  =  38°  40' 
CHECK  90°  01' 


Therefore  A  =  35°  18',  B  =  67°  24',  C  =  77°  20'. 
The  sum  of  the  half  angles  should  check  within  3'. 

107.  Area  of  a  Triangle.  It  is  shown  in  plane  geometry 
that  the  area  of  a  triangle  is  equal  to 
one  half  the  product  of  any  side  and 
the  perpendicular  from  the  opposite  ver- 
tex upon  that  side. 

If  two  sides  and  their  included  angle 
are  given,  say  b,  c,  and  A,  then 

p  =  6  sin  A 


FIG.  59 


and 

(48)  Area  =  \bc  sin  A, 

whence,  the  area  of  a  triangle  is  equal  to  one  half  the  product  of 
any  two  sides  and  the  sine  of  their  included  angle. 

If  the  three  sides  are  given,  a  formula  for  the  area  can  be 
deduced  from  (48)  as  follows.     From  (26),  §  96,  we  have 

sin  A  =  2  sin  |A  cos  %A 

9  Vs(.9  —  a)(s  —  b)(s  —  c) 
~bc~ 


V,  §  107]  TRIGONOMETRY 

by  (43)  and  (44),  §  106.     It  follows  that 
(49) 


135 


Area  =  Vs(s  —  a)(s  —  V)(s  —  c), 

in  which  s  denotes  one  half  the  perimeter. 

Let  r  be  the  radius  of  the  inscribed  circle  of  the  triangle 
whose  sides  are  a,  b,  c.     Then  since  the  area  of  the  triangle 


FIG.  60 

ABC  is  equal  to  the  sum  of  the  areas  of  the  triangles  AOB, 

BOG,  CO  A,  we  have, 

(50)  Area  =  \cr  +  \ar  +  \br  =  rs. 


Equating  (49)  and  (50),  and  dividing  through  by  s, 
(51)  r  •• 


—  c) 


which  proves  that  the  r  of  §  106  is  in  fact  the  radius  of  the  in- 
scribed circle. 

EXERCISES 
1.  Solve  each  of  the  following  triangles. 

(a)  a  =  50,  A  =  65°,  B  =  40°. 

Ans.  C  =  75°,  6  =  35.46,  c  =  53.29 

(b)  a  =  30,  b  =  54,  C  =  46°. 

Ans.  A  =  33°  6',  B  =  100°  54',  c  =  39.56 

(c)  a  =  872.5,  b  =  632.7,  C  =  80°. 

Ans.  A  =  60°  36',  B  =  39°  24',  c  =  986.2 


136 


MATHEMATICS 


[V,  §  107 


(d)  a  =  120,  b  =  80,  B  =  35°  18'. 


(a)    A 

=  21°  30', 

(b)    A 

=  62°  15', 

(c)    A 

=  53°  25', 

(d)     a 

=  30, 

(e)     a 

=  25.8, 

(/)      a 

=  37, 

(fiO     a 

=  25.3, 

(h)     a 

=  42, 

(»)      a 

=  3, 

(j)      a 

=  640, 

(fc)     a 

=  .0428, 

(0      a 

=  12, 

(m)    a 

=  6.02, 

(a)     C 

=  83°  30', 

(&)  c 

=  69°, 

(c)     C 

=  56°, 

(d)  Ci 

=  125°  14', 

C2 

=  14°  46', 

(e)      c 

=  30.57 

(/)     No  solution 

(gr)    No  solution 

(A)    B 

=  56°, 

(0    A 

=  111°  44', 

(?)    A 

=  51°  58', 

(/c)    A 

=  30°  58', 

(0    A 

=  32°  10', 

(m)  A 

=  47°  24', 

Ans.   A  =  60°,  C  = 

84°  42', 

c  =  137.9 

39,  C  =  72°  15'. 

Ans.   A  =  51°  15',  B  = 

56°  30', 

c  =  95.24 

following  triangles. 

Given  parts. 

B  =  75°, 

a  = 

31.24 

B  =  48°  45' 

6  = 

402.3 

B  =  70°  35', 

c  = 

6.031 

6  =  50, 

A  = 

20°. 

b  =  40, 

A  = 

40°  10'. 

b  =  25, 

A  = 

37°. 

6  =  54, 

A  = 

28°. 

b  =  42, 

A  - 

56°. 

b  =2, 

C  = 

30°. 

6  -  800, 

C  = 

48°  10'. 

c  =  .0832, 

B  = 

58°  30'. 

6  =  16, 

c  = 

22. 

&  =  4.82, 

c  = 

8.12 

Answers  :  Required  parts 

b  =  82.32, 

c  = 

84.68  ' 

a  =  473.4, 

c  — 

499.4 

a  =  5.841, 

6  = 

6.861 

Bl  =  34°  46', 

Ci   = 

71.63 

B2  =  145°  14', 

C2   = 

1.577 

C  =  50°, 

B  = 

90°. 

C  =  68°, 

c  = 

46.97 

B  =  38°  16', 

c  — 

2.403 

B  =  79°  52', 

c  = 

605.4 

C  =  90°  32', 

b  = 

.0709 

B  =  45°  12', 

/-Y     

102°  38'. 

B  =  36°  8', 

C  = 

96°  24'. 

V,  §  107] 


TRIGONOMETRY 


137 


3.   Find  the  areas  of  each  of  the  following  triangles. 


(a)  Given  a  =  40, 
(6)  Given  a  =  502, 

(c)  Given  a  =  27.2, 

(d)  Given  a  =  38, 


FIG.  61 


b  =  13,        c  =  37.  Ans.   Area  =  240. 

b  =  62,        c  =  484.  Ans.   Area  =  14,590. 

b  =  32.8,     C  =  65°  30'.       Ans.   Area  =  406. 
c  =  61.2,     5  =  6°  56'.      Ans.   Area  =  1,078. 

4.  Venus  is  nearer  to  the  Sun  than  the  Earth.     Assume  that  the 
orbit  of  Venus  is  a  circle  with  the  Sun  at  its  center.     The  distance  from 
the  Earth  to  the  Sun  is  92.9  millions  of  miles.     What  is  the  distance 
from  Venus  to  the  Sun  if  the  greatest  angular  distance  of  Venus  from 
the  sun  as  seen  from  the  Earth  is  46°  20'?  Ans.   67,200,000  mi. 

5.  On  a  clear  day,  twilight  ceases  when  the  sun 
has  reached  a  position  18°  below  the  horizon 
(HAS  =  18°) .     Find  the  height  AE  of  the  atmos- 
phere which  is  sufficiently  dense  to  reflect  the 
sun's  rays.     Take  OC  =  4,000  miles.     The  result 
must  be  diminished  by  20%  to  allow  for  re- 
fraction.    [MORITZ]  .  Ans.   40  miles. 

6.  The  mean  distances  of  the  Earth  and  Mars  from  the  sun  are  92.9 
and  141.5  millions  of  miles  respectively.     How  far  is  Mars  from  the 
Earth  when  its  angular  distance  from  the  sun  is  28°  10'  ? 

Ans.    21,280,000  mi. 

7.  From  two  points  on  the  same  meridian,  the 
zenith  distances  of  the  moon  are  35°  25'  and 
40°  11'.  The  difference  in  latitude  between  the 
points  of  observation  is  74°  26'.  Find  the  dis- 
tance of  the  moon  from  the  earth,  assuming  the 
FIG.  62  radius  of  the  earth  as  3,959  miles.  [MORITZ] 

Ans.   239,000  miles,  approximately. 

8.  A  search  light  20  feet  above  the  edge  of  a  tank  is  directed  to  a 
point  on  the  surface  of  the  water  40  feet  from  the  edge.     If  the  tank 
is  15  feet  deep  how  far  will  be  the  illuminated  spot  on  the  floor  of  the 
tank  from  the  edge,  the  index  of  refraction  being  4/3?    Ans.   62.5  ft. 

9.  A  man  whose  eye  is  6  feet  above  the  edge  of  a  tank  10  feet  deep  sees 
a  coin  in  a  direction  making  an  angle  of  34°  with  the  surface  of  the 
water.     If  the  index  of  refraction  is  4/3,  how  far  is  the  coin  from  the 
side  of  the  tank?  Ans.    16.83  ft. 


138  MATHEMATICS  [V,  §  107 

10.  Three  forces  of  12,  16,  and  22  pounds  in  equilibrium  can  be 
represented  by  the  3  sides  of  a  triangle  taken  in  order.     Find  the 
angles  which  they  make  with  each  other. 

Am.   77°  22',  134°  48',  147°  50'. 

11.  A  sharpshooter  and  an  enemy  are  220  feet  apart  and  on  the 
same  side  of  a  street  100  feet  wide.     Both  are  concealed  by  buildings. 
A  bullet  striking  a  building  on  the  opposite  side  of  the  street  at  an  angle 
x  is  deflected  from  the  building  at  an  angle  y  so  that  3  sin  a;  =  4  sin  y. 
Find  x  so  that  the  sharpshooter  may  be  able  to  hit  the  enemy. 

Ans.   40°  6'. 

12.  A  ship  is  going  15  miles  per  hour.     How  far  to  the  side  of  a  target 
1  mile  distant  must  the  gunner  aim  if  the  shot  travels  2000  ft.  per 
second  and  the  shot  is  fired  when  directly  opposite? 

Ans.   0°  38'  or  58  ft. 

13.  An  aeroplane  is  observed  from  the  base  and  from  the  top  of  a 
tower  40  feet  high.     The  angles  of  elevation  are  found  to  be  10°  40' 
and  9°  50'.     Find  the  distance  from  the  base  to  the  plane  and  the 
height  of  the  plane.  Ans.   2713  ft.,  502.4  ft. 

14.  To  determine  the  distance  of  a  hostile  fort  A  from  a  place  B,  a 
line  BC  and  the  angles  ABC  and  BCA  were  measured  and  found  to  be 
1006.6  yd.,  44°,  and  70°,  respectively.     Find  the  distance  AB. 

Ans.    1,036  yd. 

15.  In  order  to  find  the  distance  between  two  objects,  A  and  B, 
separated  by  a  pond,  a  station  C  was  chosen,  and  the  distance  CA 
=  426  yd.,  CB  =  322.4  yd.,  together  with  the  angle  ACB  =  68°  42', 
were  measured.     Find  the  distance  from  A  to  B.         Ans.    430.9  yd. 

16.  A  surveyor  wished  to  find  the  distance  of  an  inaccessible  point 
0  from  each  of  two  points  A  and  B,  but  had  no  instrument  with  which 
to  measure  angles.     He  measured  A  A'  =  150  ft.  in  a  straight  line  with 
OA,  and  BE'  —  250  ft.  in  a  straight  line  with  OB.     He  then  measured 
AB  =  279.5    ft.,    BA'  =  315.8    ft,,    A'B'  =  498.7  ft.       From    these 
measurements  find  each  of  the  distances  AO  and  BO. 

Ans.   152.3  ft.,  319.7  ft. 

17.  Two  stations,  A  and  B,  on  opposite  sides  of  a  mountain,  are  both 
visible  from  a  third  station  C.     The  distance  AC  =  11.5  mi.,  BC  =  9.4 
mi.,  and  angle  ACB  =  59°  30'.     Find  the  distance  between  A  and  B. 

Ans.    10.5  mi. 


CHAPTER  VI 


LAND   SURVEYING 

108.  The  Surveyor's  Function.     Land   surveying   consists 
in  measuring  distances  and  angles  and  marking  corners  and  lines 
upon  the  ground,  and  in  recording  these  measurements  in  field 
notes  from  which  a  map  can  be  drawn  and  the  area  computed. 

The  original  survey  of  a  tract  of  land  having  been  made  and 
recorded,  a  surveyor  may  subsequently  be  called  upon  to  find 
the  corners,  to  relocate  them  if  lost,  to  retrace  the  old  boundaries, 
and  to  renew  the  corner  posts  and  monuments  if  decayed  or 
destroyed.  This  is  called  a  resurvey. 

A  surveyor  may  make  a  resurvey  of  a  tract  of  land  in  order 
to  divide  it  by  new  lines  and  to 
map   and  compute  the  areas  of 
the  subdivisions. 

109.  Instruments.    Distances 
on  the  ground  are  measured  with 
the  chain  or  tape.     The  land  sur- 
veyor's chain  is  66   feet  (4  rods) 
long  and  is  divided  into  100  links 
each  7.92  inches  long.     The  steel 
tape  is  usually  100  feet  long,  sub- 
divided to  hundredths  of  a  foot. 

Angles,  horizontal  or  vertical, 
are  usually  measured  with  the 
transit.  This  is  an  instrument 

mounted  on  a  tripod,  and  composed  of  the  following  parts:  (a) 
the  telescope  provided  with  cross  hairs  to  determine  the  line  of 
sight,  a  sensitive  spirit  level,  and  a  graduated  circle  on  which  the 

139 


FIG.  63 


'  140  MATHEMATICS  [VI,  §  109 

angular  turn  of  the  telescope  in  the  vertical  plane  is  read;  (6) 
the  alidade,  carrying  the  telescope,  provided  with  spirit  levels 
to  bring  its  base  into  the  horizontal  plane  and  a  large  gradu- 
ated circle  on  which  is  read  the  angular  turn  of  the  telescope  in 
measuring  horizontal  angles;  and  (c)  the  magnetic  compass. 

110.  Bearing  of  Lines.     The  direction  of  a  line  on  the  ground 
may  be  given  by  its  bearing;  this  is  the  angle  between  the  line 
and  the  meridian  through  one  end  of  it.     For  example,  a  line 
bearing  N  26°  E  is  one  which  makes  an  angle  of  26°  on  the  east 
side  of  north;  one  bearing  S  85°  W  makes  an  angle  of  85°  on  the 
west  side  of  south.     The  bearing  of  a  line  which  is  run  by  the 
transit  is  read  off  on  the  compass  circle  but  is  subject  to  a  cor- 
rection depending  upon  the  time  and  place  since  the  magnetic 
needle  does  not  point  due  north  at  all  times  and  places. 

111.  Government  Surveys.     In  government  surveys  of  the 
public  lands,  a  north  and  south  line  called  a  principal  meridian 
is  first  accurately  laid  out  and  marked  by  permanent  monuments. 
From  a  convenient  point  on  the  principal  meridian  a  base  line 
is  run  east  and  west  and  carefully  marked.     North  and  south 
lines,  called  range  lines,  are  then  run  from  points  six  miles  apart 
on  the  base  line.     Then  township  lines  six  miles  apart  are  run 
east  and  west  from  the  principal  meridian. 

The  land  is  thus  divided  into  townships  six  miles  square.  A 
tier  of  townships  running  north  and  south  is  called  a  range. 
Ranges  are  numbered  consecutively  east  and  west  from  the 
principal  meridian.  Townships  are  numbered  north  and  south 
from  the  base  line. 

In  deeds  and  records  a  township  is  located,  not  by  the  county, 
but  as  "  Township  No.  —  north  (or  south)  of  a  certain  base  line 
and  in  range  No.  —  east  (or  west)  of  a  certain  principal  me- 
ridian. Townships  are  divided  into  thirty-six  sections  each 
one  mile  square  containing  640  acres,  and  are  numbered  from 


VI,  §  111] 


LAND   SURVEYING 


141 


1  to  36  as  shown  in  Fig.  64.     The  sections  are  often  subdivided 
into  halves,  quarters,  eighths,  etc.,  as  illustrated  in  Fig.  65. 


6 

S 

4 

3 

2 

1 

7 

8 

9 

10 

11 

12 

18 

17 

16 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

Iff  A. 

NE  1 

s.j-x.w.j 

160  A. 

80  A. 

t 

I 

2  , 

s.w.4 

S~E.  i 

4 

160  A. 

80  A. 

FIG.  64 


FIG.  65 


The  first  principal  meridian  runs  north  from  the  junction  of 
the  Ohio  and  Big  Miami  rivers  on  the  boundary  between  Ohio 
and  Indiana.  The  second  coincides  with  86°  28'  of  longitude 
west  of  Greenwich  running  north  from  the  Ohio  river  near  the 
towns  of  English,  Bedford,  Lebanon,  Culver,  Walkerton,  and 
Warwick,  Indiana.  The  surveys  in  Indiana  (with  the  exception 
of  certain  lands  in  the  southeast  corner)  are  governed  by  this 
second  principal  meridian  and  a  base  line  in  latitude  38°  28'  20" 
crossing  this  meridian  about  5  miles  south  of  Paoli,  in  Orange 
County.*  Thus  a  certain  parcel  of  land  is  described  in  the 
Indiana  records  as  "  E  \  of  NW  \  of  Section  nineteen  (19), 
Township  twenty-three  (23)  N,  Range  four  (4)  W." 

The  surveys  extending  east  from  one  meridian  will  not  gener- 
ally close  with  those  extending  west  from  the  preceding  meridian; 
the  same  is  true  of  the  ranges  of  townships  extending  north 

*  The  first  six  principal  meridians  are  designated  by  number ;  some  twenty -odd  others 
by  name.  E.  g.,  the  Mount  Diablo  meiidian,  120°  54'  48"  W,  which  governs  sur- 
veys in  California  and  Nevada.  The  first  six  base  lines  are  neither  numbered  nor 
named  but  all  subsequent  ones  are  named.  The  locations  of  all  the  principal  merid- 
ians and  base  lines  is  given  in  the  Manual  of  Instructions  for  the  Survey  of  the  Public  Lands 
issued  from  timo  to  time  by  the  GENERAL  LAND  OFFICE,  Washington.  D.  C.  For  de- 
tails and  a  historical  sketch  see  also,  PENCE  AND  KETCHUM,  Surveying  Manual. 


142  MATHEMATICS  [VI,  §  111 

and  south  from  the  base  lines.  These  circumstances  and  the 
presence  of  rivers  and  lakes  give  rise  to  fractional  townships  and 
sections. 

112.  Corners.     In  an  original  survey  one  of  the  most  im- 
portant of  the  surveyor's  duties  is  the  marking  of  corners  in 
such  a  manner  as  to  perpetuate  their  location  as  long  as  possible. 
The  Manual  of  Instructions  (see  1894  edition,  p.  44)  says,  "  If 
the  corners  be  not  perpetuated  in  a  permanent  and  workman- 
like manner,  the  principal  object  of  the  surveying  operations 
will  not  have  been  attained." 

The  Instructions  prescribe  in  detail  the  kind  of  monument  and 
the  mark  to  be  put  upon  it  to  establish  each  of  the  various  kinds 
of  corners  that  are  located  in  the  government  surveys.  Wooden 
posts  and  stakes,  stones,  trees,  and  mounds  of  earth  are  used. 
Witness  trees  or  witness  points  are  trees  or  other  objects  located 
near  the  corner,  suitably  marked  and  described  in  the  field  notes 
to  make  easy  a  subsequent  relocation  of  the  corner. 

If  called  upon  to  make  a  resurvey  of  land  that  was  originally 
laid  out  under  the  direction  of  the  General  Land  Office,  the 
surveyor  will  do  well  to  make  a  careful  study  of  the  instructions 
concerning  corners  that  were  in  force  when  the  original  survey 
was  made,  as  the  practice  has  varied  somewhat  from  time  to 
time. 

113.  Judicial  Functions  of  Surveyors.*    Many  years  have 
elapsed  since  the  greater  part  of  the  government  surveys  were 
made  and  in  many  cases  the  original  corner  marks  have  entirely 
disappeared.     The  first  settlers  and  original  owners  often  failed 
to  fix  their  lines  accurately  while  the  monuments  remained,  and 
the  subsequent  owners  have  no  first  hand  knowledge  of  their 
location.     When  in  such  cases  a  surveyor  is  called  upon  to 

*  This  topic  is  based  upon  a  paper  of  the  same  title  by  Justice  Cooley  of  the  Michi- 
gan Supreme  Court,  published  in  the  Michigan  Engineer's  Annual  for  1880-81. 
pp.  18-25. 


VI,  §  114]  LAND  SURVEYING 


1* 


make  a  resurvey,  it  is  his  duty  to  find  if  possible  where  the 
original  corners  and  boundary  lines  were,  and  not  at  all  where 
they  ought  to  have  been.  However  erroneous  the  original  sur- 
vey may  have  been,  the  monuments  that  were  set  must  never-^ 
theless  govern,  for  the  parties  concerned  have  bought  with  refer- 
ence to  these  monuments  and  are  entitled  only  to  what  is 
contained  within  the  original  lines. 

If  the  original  monument  and  all  the  witness  trees  and  other 
identification  marks  mentioned  in  the  field  notes  of  the  original 
survey  have  disappeared,  the  corner  is  lost  and  it  is  the  duty  of 
the  surveyor  to  relocate  it  in  the  light  of  all  the  evidence  in  the 
case,  including  the  testimony  of  persons  familiar  with  the 
premises,  existing  fences,  ditches,  etc.,  at  the  point  where  this 
evidence  shows  it  most  probably  was. 

In  making  a  resurvey  the  surveyor  has  no  authority  to 
settle  disputed  points ;  if  the  disputing  parties  do  not  agree 
to  accept  his  decision,  the  question  must  be  settled  in  the 
courts.  In  a  controversy  between  adjacent  owners  over  the 
location  of  corners  and  division  lines,  it  is  well  established  in  law 
that  a  supposed  boundary  line  long  accepted  and  acquiesced 
in  by  both  parties  is  better  evidence  of  where  the  real  line  should 
be  than  any  survey  made  after  the  original  monuments  have 
disappeared.  It  is  common  belief  that  boundary  lines  do  not 
become  fixed  by  acquiescence  in  less  than  21  years,  but  there  is 
no  particular  time  that  must  elapse  to  establish  boundary  lines 
between  private  owners  where  it  appears  that  they  have  ac- 
cepted a  particular  line  as  their  boundary  and  all  concerned 
have  claimed  and  occupied  up  to  it. 

114.  Measuring  on  Level  Ground.  The  line  to  be  meas- 
ured is  first  ranged  out  and  marked  with  range  poles  or  its 
direction  is  determined  by  the  line  of  sight  of  the  transit 
set  on  the  line.  The  leader  sticks  a  pin  at  the  starting  point, 
takes  ten  in  his  hand  and  steps  forward  on  the  line  dragging  the 


44  MATHEMATICS  [VI,  §  114 

chain  behind  him.  At  a  signal  from  the  follower,  given  just 
before  the  full  length  has  been  drawn  out,  he  turns,  aligns,  and 
levels  the  chain,  stretches  it  to  the  proper  tension,  and,  while 
the  follower  holds  the  rear  end  at  the  starting  point,  sticks  a  pin 
at  the  forward  end  on  the  line  determined  by  the  follower  and 
a  range  pole  or  by  the  transitman.  At  a  signal  from  the 
leader  the  follower  pulls  his  pin  and  both  move  forward  on  the 
line  another  chain's  length  and  set  the  next  pin.  This  process 
is  repeated  until  the  leader  has  set  his  tenth  pin,  when  the 
follower  goes  forward,  counting  his  pins  as  he  goes  and,  if  there 
are  ten,  hands  them  to  the  leader  who  also  counts  them.  The 
count  of  tallies  is  kept  by  both.  When  the  end  of  the  line  is 
reached  the  follower  walks  forward  and  reads  the  fraction  of 
the  chain  at  the  pin  and  notes  the  number  of  pins  in  his  hand  to 
determine  the  distance  from  the  last  tally  point  which  is  re- 
corded in  the  field  notes. 

115.  Measuring  on  Slopes.     The  horizontal  distance  which 
is  required  can  be  found  on  slopes  by  leveling  the  chain  and 

plumbing  down  from  the  end 
off  ground.  On  steep  slopes 
only  a  part  of  the  chain  can 
be  used  as  at  A  and  B  in  Fig. 
66.  The  part  used  should  be 
a  multiple  of  ten  links  and 
great  caution  must  be  used 

by  both  leader  and  follower  to  avoid  mistakes  and  confusion  in 

the  count  of  pins. 

116.  Offsets.     In  case  measurements  cannot  be  made  on  the 
desired  line  on  account  of  a  fence,  hedge,  pond  or  other  obstacle, 
a  perpendicular  to  the  line,  called  an  offset,  is  measured,  suf- 
ficiently long  to  avoid  the  obstruction  and  the  measurements  are 


VI,  §  117] 


LAND  SURVEYING 


145 


made  on  an  auxiliary  line  parallel  to  the  required  line.  Stakes 
may  then  be  set  on  the  required  line  by  offsets  from  the  auxiliary 
line.  See  Fig.  67. 

c  D 


FIG.  68 


FIG.  67 

From  any  point  on  a  line  a  right  angle  (or  any  other  required 
angle)  can  be  laid  off  with  the  tran- 
sit. An  angle  of  90°  or  60°  can  be 
laid  off  in  a  clear  space  with  chain  or 
tape  and  pins  as  shown  in  Fig.  68, 
from  the  facts  that  (1)  a  triangle 
whose  sides  are  to  each  other  as  3  : 
4  :  5  has  a  right  angle  opposite  the 
longest  side;  and  (2)  an  equilateral  triangle  has  three  60°  angles. 

117.  Passing  Obstacles.  An  obstacle  in  the  line  can  be 
passed  and  the  line  prolonged  beyond  it  by  means  of  perpen- 
dicular offsets  as  shown  in  Fig.  67,  if  the  nature  of  the  locality 
makes  it  convenient. 

The  same  result  can  be  accomplished  by  a  triangle  as  shown 
in  Fig.  69.  The  angle  HAB,  the  distance  AB,  the  angle  KBC, 

are  measured;  then  the  distance  BC 
and  the  angle  MOD  are  computed; 
the  distance  BC  is  measured  off  and 
the  point  C  is  located  and  the  angle 
at  C  is  turned  off  and  the  direction 
CD  established;  AC  is  computed 
and  the  point  D  is  taken  a  whole 
The  angles  at  A  and  B  and  the 


FIG.  69 


number  of  chains  from  A 
11 


146  MATHEMATICS  [VI,  §117 

distance  AB  are  arbitrary  and  may  be  taken  so  as  to  avoid 
difficulties  of  the  surroundings.  If  the  circumstances  permit 
the  angle  HAB  may  be  made  60°,  and  angle  KEG  =  120°; 
then  the  triangle  ABC  will  be  equilateral  and  computations  will 
be  avoided. 

1 18.  Random  Lines.     When  it  is  desired  to  mark  out  a  long 
line,  such  as  AB,  Fig.  70,  whose  end  points  are  established  but 

A^  k  T&  Ts  Tt  T5         B 

FIG.  70 

are  invisible  each  from  the  other,  a  line  AC,  called  a  random  line, 
is  run  as  nearly  in  the  direction  of  A  B  as  can  be  determined 
and  stakes  Si,  S2,  83,  etc.,  are  set  at  regular  measured  distances. 
On  coming  out  near  B  a  perpendicular  is  let  fall  from  B  to  AC 
precisely  locating  the  point  C.  The  lengths  of  the  offsets 
SiTi,  $2^2,  $3^3,  etc.,  all  perpendicular  to  AC,  can  be  computed 
and  on  retracing  CA,  stakes  can  be  set  at  T$,  Tt,  T3,  etc.,  on 
the  desired  line  AB.  For  example,  if  the  stakes  on  AC  are  12 
chains  apart,  if  S$C  =  6.46  chains,  and  if  BC  =  54  links,  then 
the  offset,  in  links,  at  any  stake  S,  is  found  by  multiplying  its 
distance  AS,  in  chains,  from  A,  by  the  ratio  54/66.46  =  0.8125. 
Thus  the  offset  S4T4  =  48  X  0.8125  =  39.  It  is  left  to  the 
student  to  show  that  A  B  is  longer  than  AC  by  less  than  1/4  a 
link  and  that  the  stakes  on  AB  are  practically  12  chains  apart. 

119.  Computing  Areas.     If  the  boundaries  of  a  tract  are  all 
straight  lines,  i.  e.,  if  its  perimeter  is  a  polygon,  the  area  can  be 
computed  by  dividing  it  into  triangles,  or  into  rectangles  and 
triangles,  provided  enough  measurements  are  made  so  that  the 
required  dimensions  of  each  part  are  known  or  can  be  com- 


VI,  §121] 


LAND   SURVEYING 


147 


puted.  It  is  customary  to  measure  more  lines  on  the  ground 
than  is  theoretically  necessary  in  order  to  check  the  computa- 
tions. These  extra  measurements  are  called  proof  lines  in  the 
field  notes. 

120.  Irregular  Areas  by  Offsets.     When  one  side  of  a  field 
is  not  straight  as  occurs  if  the  boundary  is  a  stream  or  curved 
road,  a  line  may  be  run  cutting  off  the 

irregular  part  and  leaving  the  remain- 

der of  the  field  in  a  shape  whose  area 

is  easily  computed;  as  AD  in  Fig.  71. 

Stakes  are  set  at  regular   measured 

intervals  on  AD  and  the  offsets  AB, 

SiTi,  SzTz,  SzTs,  etc.,  are  measured. 

The  area  can  be  approximated  by  considering  each  of  the  strips 

to  be  a  trapezoid.     On  computing  and  adding  we  are  led  to  the 

following  rule. 

RULE:  From  the  sum  of  all  the  offsets  subtract  half  the  sum  of 
the  extreme  ones  and  multiply  the  remainder  by  the  common 
distance  between  them. 

121.  Areas  by  Rectangular  Coordinates.     If  the  irregular 

side  of  the  field  is  a  broken 
line  or  if  the  nature  of  the 
place  makes  it  inconvenient 
to  measure  the  offsets  at  regu- 
lar intervals  the  area  can  be 
found  by  measuring  the  rec- 
tangular coordinates  of  the 

points  A,  B,  C,  D,  E,  Fig.  72,  referred  to  the  axes  OX  and  OY. 
Let  the  coordinates  of  A,  B,  C,  •  •  •  be  (0,  y0),  (zi,  T/I),  (x2,  7/2), 

•  •  •  respectively.     Then  the  sum  of  the  areas  of  the  trapezoids  is 


FIG.  72 


(1) 


2/2) 


148 


MATHEMATICS 


[VI,  §  121 


where  n  is  the  number  of  trapezoids.     On  combining  terms 
this  reduces  to 

(2)  |[{zi(ffe  -  2/2)  +  £2(2/1  -  2/3)  +  •  •  • 

+  £n_l(2/n-2   -  2/n)}    +  £n(2/n-l  +  2/n)]- 

Hence  we  have  the  following  rule. 

RULE:  From    each    ordinate    subtract    the    second    succeeding 

ordinate  and  multiply  the  remainder  by  the  abscissa  of  the  inter- 

mediate point;  also  multiply  the  sum  of  the  last  two  ordinates  by 

the  last  abscissa;  and  divide  the 
algebraic  sum  of  the  products  by 
two. 

If  the  coordinates  of  the  ver- 
tices of  a  closed  polygon  are 
known  its  area  can  be  computed 
as  follows.  Consider  the  con- 
vex pentagon  shown  in  Fig.  73. 
The  area  included  may  be  found 
by  adding  the  trapezoids  under 

the  sides  ED  and  DC  and  subtracting  those  under  the  other 

three  sides;  this  gives 

(3)  |[(z4  -  £5X2/4  +  2/5)  +  (£3  -  £4X2/3  +  2/4) 

—  (£3  —  £2X2/3  +  2/2)  —  (£2  —  £1X2/2  +  2/0 

-  (xi  -  x5)(yi  +  2/5)]., 

Combining  like  terms,  we  find  that  this  reduces  to  either 

(4)  $[xi(yt  -  2/5)  +  £2(2/3  -  2/i)  +  £3(2/4  -  2/2) 

+  £4(2/5  -  2/s)  +  £5(2/1  -  2/4)], 


TJI       ~o 


or 


(5)    ?[yi(x6  —  £2)  +  yz(x\  —  £3)  +  2/3(^2  —  £4) 

—  £5) 


—  £1)]. 


VI,  §  121] 


LAND  SURVEYING 


149 


These  formulas  are  easily  extended  to  convex  polygons  of 
any  number  of  sides  and  prove  the  following  rule. 

Multiply  each  abscissa  by  the  difference  of  its  adjacent  ordinatcs, 
always  making  the  subtractions  in  the  same  sense  around  the  perim- 
eter, and  take  one-half  the  algebraic  sum  of  the  products. 

The  result  will  be  the  same  (except  as  to  sign)  if  in  this  rule 
the  words  abscissa  and  ordinate  be  interchanged. 

EXERCISES 

1.  Find  the  area  of  a  field  in  the  form  of  a  right  triangle. 

(a)   Base  =  31.28  ch.,     Altitude  =  16.25  ch.  Ans.   25.42  A. 

(6)   Base  =  28.46  ch.,    Altitude  =  38.65  ch.  Ans.  55.00  A. 

2.  Find  the  area  of  a  triangular  field, 

(«)   whose  three  sides  are  24.50,  10.40,  and  21.70  ch. 
(6)   having  two  sides  35.60,  23.70  ch.,  and  their  included  angle  42°  30'. 

Ans.   (a)    11.27  A.     (6)   28.50  A. 

3.  How  many  acres  in  a  rectangular  field  whose  dimensions  are 
17.44  and  32.65  ch.  Ans.  56.94  A. 

4.  One  side  of  a  200  acre  rectangular  field  is  33.60  chains.     Find  the 
other  side.  Ans.   62.50  ch. 

5.  What  is  the  length  of  one  side  of  a  square  field  which  contains 
36  acres?  Ans.   18.97  ch. 

6.  The  diagonals  of  a  four-sided  field  measure  21.40  and  24.50  ch., 
and  they  cross  at  an  angle  of  74°  40'.     Find  the  area.     Ans.   25.28  A. 

7.  One  diagonal  of  a  quadrangle  runs  N.  36°  20'  E.  22.40  ch.,  and  the 
other  S.  69°  30'  E.     Find  its  area.  Ans.   25.22  A. 

8.  Find  the  areas  of  the  fields  whose  boundaries  are  given. 


Sta- 
tion. 

Bearing. 

Distance. 

A 

North 

9.14  ch. 

B 

S.  73°  25'  E. 

8.27 

C 

S.  28°  15'  E. 

10.04 

D 

N.  80°  45'  W. 

12.84J 

Sta- 
tion. 

Bearing. 

Distance. 

P 

West 

19.66  ch. 

Q 

North 

13.77 

R 

N.  64°  15'  E. 

16.66 

S 

S.  12°  30'  E. 

21.51 

Ans.    (a)  8.74  A.     (6)  30.97  A. 


150 


MATHEMATICS 


[VI,  §  121 


9.    Find  the  areas  of  the  fields  whose  boundaries  are  given, 
(a)  (6) 


Sta- 
tion. 

Bearing. 

Distance. 

A 

N.  25°  30'  E. 

10.50  ch. 

B 

N.  76°  50'  E. 

7.00 

C 

S.  19°  30'  E. 

7.92 

D 

S.  53°  34'  W. 

11.90 

E 

N.  64°  30'  W. 

4.20 

Sta- 
tion. 

Bearing. 

Distance. 

1.  . 

N.  12°  46'  W. 

6.80  ch. 

2.  . 

N.  49°  10'  E. 

2.40 

3.  . 

S.  40°  50'  E. 

6.00 

4.  . 

S.  10°  30'  W. 

4.00 

5.  . 

N.  85°  50'  W 

4.52| 

Ana.    (a)  10.09  A.     (6)  3.30  A. 
10.  The  coordinates,  in  chains,  of  the  vertices  of  a  broken  line  are: 


Vertex. 

A. 

B. 

c. 

D. 

E. 

F. 

X   

0.00 

2.95 

1.10 

0.60 

2.20 

1.80 

y  

10.00 

8.12 

7.25 

5.00 

4.50 

0.00 

Find  the  area  included  by  the  broken  line  and  the  axes.     Ans.  2.36  A. 
11.  The  coordinates,  in  chains,  of  the  corners  of  a  field  are: 


Vertex. 

1. 

2. 

3. 

4. 

5. 

6. 

X    

0.00 

7.00 

12.50 

18.00 

15.00 

10.00 

y   

6.00 

12.00 

20.00 

15.00 

8.25 

0.00 

Make  a  plot  and  find  the  area. 


Ans.  16.175  A. 


12.  Starting  on  the  bank  of  a  river  a  line  is  run  across  a  bend  20.00 
ch.,  to  the  bank  again.     Offsets  are  measured  every  two  chains  as  fol- 
lows: 1.61,  2.27,  1.96,  4.23,  3.70,  2.92,  3.26,  2.50,  1.25  ch.     Make  a 
plot  of  the  land  between  the  line  and  the  river  and  find  the  area. 

Ans.  4.74  A. 

13.  Find  the  measurements  so  as  to  run  a  line  from  the  vertex  A 
of  a  triangle  ABC  to  a  point  D  on  the  side  BC  =  8.75  ch.,  so  as  to  cut 
off  2/5  of  the  area  next  to  B.  Ans.  BD  =  3.50  ch. 

14.  Find  the  measurements  so  as  to  run  a  line  through  a  point  E 
on  BC  of  the  triangle  of  Ex.  13,  parallel  to  AB  so  as  to  cut  off  2/5  of 
the  area  in  the  trapezoid.  Ans.  CE  =  6.78  ch. 


VI,   §  121] 


LAND   SURVEYING 


151 


15.  Two  lines  meet  at  P.     PA  bears  S.  65°  30'  E.,  PB  bears  N.  78° 
15'  E.     Determine  measurements  to  run  a  line  perpendicular  to  PA 
so  as  to  cut  off  five  acres.     Ans.  Base  =  lOVctn  36°  15'  =  11.68  ch. 

16.  A  triangular  field  contains  6  A.     Show  how  to  find  on  the  plot 
a  point  inside  the  triangle  from  which  lines  drawn  to  the  vertices  will 
divide  it  into  three  triangular  fields  of  1,  2,  and  3  A.,  and  so  that  the 
smallest  and  largest  shall  be  adja- 
cent respectively  to  the  smallest  and 

largest  sides  of  the  field. 

17.  If  the   bases  of  a  trapezoid 
are  a  and  b,  a  <  6,  and  the  slant 

sides  are  c  and  d,  as  in  Fig.  74,  de-  FIG.  74 

termine  measurements  to  run  a  line 

parallel  to  the  bases  to  cut  off,  adjacent  to  the  shorter  base  a,  a  frac- 
tion /,  of  the  whole  area. 


Ans.  x  =  Va2  +  /(b2  -  a2),     y  =  c 


b  -a' 


b  -a' 


18.  Given  a  =  20,  b  —  30,  c  =  54.40  ch.,  determine  x  and  y  to  cut 
off  \  the  area,  Fig.  74.  Ans.  x  =  23.80,     y  =  20.69  ch. 

19.  In  a  four  sided  field  ABCD,  AB  runs  S.  8.40  ch.,  BC,  E.  9.24  ch., 
and  CD,  N.  5.68  ch. 

(a)  Run  a  north  and  south  line  so  as  to  divide  it  into  two  parts  whose 
areas  shall  be  to  each  other  as  2  :  3  with  the  smaller  on  the  east. 

(b)  Run  a  north  and  south  line  so  as  to  cut  off  3  A.  on  the  west. 

Ans.  (a)  4.14  ch.  from  the  east;     (6)  5.40  ch.  from  the  east. 

20.  A  tract  of  land  A  BCD,  lies 
between  two  converging  streets  as 
shown  in  Fig.  75.  AB  =  1980  ft., 
AC  =  590  ft.,  BD  =  1380  ft.  De- 
termine the  measurements  for  run- 
ning lines  PQ,  RS,  etc.,  perpen- 
dicular to  AB,  so  as  to  divide  the 
tract  into  ten  lots  of  equal  area. 

[HINT.  Use  the  method  of  Ex.  17 
to  find  PQ  and  AP.  Or  otherwise, 


1 


^^ 

S 

Q 

c 

A 

P 

R 

FIG.   75 
find  the  tangent  of  the  angle  between  the  streets  AB  and  CD ;   find  the 


MATHEMATICS 


[VI,  §  121 


Corner. 

Bearing. 

Distance. 

1.  . 

N.  75°  30'  W. 
N.  5°  15'  E. 
S.  68°  10'  E. 
S.  23°  E. 

30.08  ch. 

2   ... 

3 

4  

Area  =  139.84  acres 

area  of  CAPQ  in  terms  of  x(  =  AP) ;  this  leads  to  a  quadratic  equation 

in  x.     Find  the  positive  root.] 

Ans.    AP  =  300.11  ft.      PQ  =  709.74  ft.      Area  CAPQ  =  4.477  A. 

21.  From  the  notes  in  Ex.  8  (6),  make  a  plot  and  (a)  run  a  line  from  S 
to  a  point  M  on  PQ  so  as  to  divide  the  field  into  two  parts  of  equal  areas, 
(6)  run  a  line  from  R  to  a  point  N  on  SP  so  as  to  cut  off  10  acres  in  the 
triangle. 

22.  From   the   accompany- 
ing notes  from  a  farm  survey 
compute   the   lengths   of   the 
first,  second,  and  fourth  sides. 

[HiNT.  Produce  the  second 
and  fourth  sides  to  form  a 
triangle.]  Ans.  51.38,  36.56,  40.16. 

23.  Suppose  the  lengths  of  the  first  and  fourth  sides  of  the  field 
in  Ex.  9  (a)  are  unknown.     Compute  them  from  the  other  data  if  the 
area  is  10.094  acres. 

24.  It  is  desired  to  mark  out  and  measure  a  line  PQ.     A  random 
line  PR  is  run  and  stakes  are  set  on  it  every  100  ft.     The  perpendicular 
from  Q  upon  PR  is  48.82  ft.  long  and  meets  it  at  R,  22.18  ft.  beyond  the 
42nd  stake.     Compute  the  offsets  for  setting  the  stakes  over  on  PQ, 
their  distance  apart,  and  the  length  of  PQ. 

25.  To  prolong  a  line  AB  past  an  obstacle  0,  a  right  turn  40°  is 
made  at  B,  400  ft.  is  measured  to  C,  and  a  left  turn  of  116°  is  made. 
Compute  the  distance  to  D  on  AB  produced  through  O  and  the  right 
turn  which  must  be  made  at  D.     How  far  from  D  should  hundred  foot 
stakes  be  resumed? 


CHAPTER  VII 
STATICS 

122.  Statics.     Statics  treats  of  bodies  at  rest  and  of  bodies 
whose  motion  is  not  changing  in  direction  or  in  speed.     A  body 
whose  motion  is  not  changing  is  said  to  be  in  equilibrium.     The 
chief  problem  of  statics  is  to  find  the  conditions  of  equilibrium. 

123.  Mass.     The  weight,  W,  of  a  body  is  not  constant.     For 
instance  a  body  weighs  less  on  a  mountain  top  than  at  sea  level. 
Also  the  acceleration,  g,  due  to  gravity  is  not  constant.     It 
likewise  is  less  on  a  mountain  top  than  at  sea  level.     An  increase 
in  acceleration  is  accompanied  by  a  proportional  increase  in 
weight.     But  the  ratio  W/g  is  constant.     The  constant  number 
represented  by  this  ratio  is  called  the  mass  of  the  body.     A  unit 
of  mass  is  the  gram,  and  is  1/1000  of  the  mass  of  a  certain  piece 
of  platinum  which  is  preserved  at  Paris.     Another  unit  of  mass 
is  the  avoirdupois  pound.     One  thousand  grams  is   equal   to 
2.20462125  Ibs.     The  mass  of  any  body  is  then  the  number  ex- 
pressing the  ratio  of  its  weight  to  the  weight  of  a  unit  of  mass. 
The  weight  is  to  be  determined  by  means  of  a  spring  balance. 

124.  Momentum.     When  a  given   mass  is  in   motion,  we 
require  to  know  not  only  the  magnitude  of  the  mass,  but  also 
its  velocity.      The  product  of  the  mass  of  a  body  and  its  velocity 
is  called  its  momentum. 

125.  Force.     If  a  body  possesses  a  certain  amount  of  momen- 
tum, it  is  impossible  for  it  to  alter  its  motion  in  any  manner 
unless  acted  upon  by  some  other  body  which  pushes  or  pulls  it. 

Force  is  that  which  tends  to  produce  a  change  of  motion  in  a 
body  on  which  it  acts.  This  change  of  motion  is  proportional  to 
the  force  and  takes  place  in  the  direction  of  the  straight  line  in 

153 


154  MATHEMATICS  [VII,  §  125 

which  the  force  acts.  Thus,  to  increase  the  speed  of  an  auto- 
mobile, the  driving  force  must  be  increased.  The  greater  the 
force,  the  greater  the  rate  of  increase  in  the  speed. 

This  illustrates  the  fact  that  forces  are  of  different  magnitudes. 

If  a  motionless  croquet  ball  is  struck,  its  subsequent  motion 
depends  upon  the  direction  of  the  stroke.  This  illustrates  the 
fact  that  forces  have  different  lines  of  action.  If  a  billiard  ball 
is  struck,  the  motion  of  the  ball  depends  upon  the  point  at  which 
the  cue  struck  the  ball.  This  illustrates  the  fact  that  we  must 
take  into  account  the  point  of  application  of  the  force. 

A  force  is  said  to  be  completely  determined  if  we  know  (a)  its 
magnitude;  (6)  its  line  of  action;  (c)  its  direction  along  the 
line  of  action;  (d}  its  point  of  application. 

In  practice  forces  are  never  applied  at  a  point.  The  force  is 
applied  over  an  area  such  as  the  pressure  of  a  thumb  on  the 
head  of  a  tack  or  the  pressure  of  a  book  on  a  table.  A  force 
may  act  throughout  an  entire  volume  as  is  the  case  with  at- 
traction. These  forces  are  called  distributed  forces.  In  prac- 
tice we  often  consider  the  forces  which  applied  at  a  point  would 
produce  the  same  effect  as  the  given  distributed  forces.  Such 
forces  are  termed  concentrated  forces. 

126.  Unit  of  Force.     The  unit  of  force  is  sometimes  taken 
as  the  weight  of  a  unit  mass.     This  unit  of  force  is  not  constant. 
It  changes  both  with  altitude  and  with  latitude.     These  changes 
are  small  but  for  scientific  purposes  cannot  be  neglected.     To 
obtain  a  constant  unit  it  is  sufficient  to  make  the  following 
definition : 

The  unit  of  force  is  the  weight  of  a  unit  of  mass  at  a  fixed  place, 
say  at  London,  Paris,  or  Washington. 

127.  Graphic  Representation  of  Forces.  A  force  P  is  com- 
pletely determined  if  we  know  its  magnitude,  its  line  of  action, 
its  direction  along  this  line,  and  its  point  of  application.     It 


VII,  128J  STATICS  155 

follows  that  a  force  can  be  completely  represented  by  anything 
which  possesses  these  attributes.     It  can,  for  example,  be  repre- 
sented by  a  directed   segment  of  a 
straight  line.      For  we  may  let  any 
point  0,  Fig.  76,  represent  the  point 
of  application.      From  0  draw  any 
line  segment  OA  the  number  of  units 

in  whose  length  is  the  same  as  the  number  of  units  in  the  given 
force.  The  length  of  the  segment  represents  then  the  magnitude 
of  the  force.  The  line  of  which  OA  is  a  part  represents  the  line 
of  action  of  the  force.  We  can  represent  the  direction  along  the 
line  by  an  arrowhead  placed  on  OA. 

128.  Composition  of  Forces.  —  Parallelogram  of  Forces. 
If  two  or  more  forces  act  in  the  same  straight  line  and  in  the 
same  direction,  their  resultant,  or  sum,  is  obtained  by  adding 
the  numbers  representing  the  magnitudes  of  the  forces. 

If  the  forces  act  in  the  same  straight  line  but  in  opposite 
directions,  the  resultant  is  equal  to  their  difference,  that  is  to 
their  algebraic  sum. 

When  the  forces  do  not  act  in  the  same  straight  line  the 
total  or  resultant  force  is  found  by  means  of  a  rule  called  the 
parallelogram  of  forces:  If  two  forces  not  in  the  same  straight 
line  are  represented  in  direction  and  in  magnitude  by  two  adjacent 
sides  of  a  parallelogram,  the  single  force  which  would  produce  the 
same  effect  as  the  two  given  forces  is  represented  in  direction  and 
in  magnitude  by  that  diagonal  of  the  parallelogram  which  passes 
through  the  same  vertex  as  the  two  given  forces. 

In  Fig.  77,  the  forces  FI  and  F2  are  represented  by  the  lines 
AB  and  AC,  respectively.  Their  resultant  R  is  represented  by 
AD.  The  magnitude  of  the  resultant  is  given  by  the  equation 


(1)  R  =  VFS  +  F22  +  2FiF2  cos  0, 

where  6  stands  for  the  angle  BAC. 


156  MATHEMATICS  [VII,  §  128 

It  will  be  noted  that  BD,  being  parallel  and  equal  to  AC, 
represents  the  magnitude  and  the  direction  (but  not  line  of 


FIG.  77 

action)  of  the  force  F2.     If  we  let  a  equal  the  angle  BAD,  we 
have,  from  the  triangle  BAD 


sine      sin  a' 
whence 

F2  sin  9 
(3)  sin  a  = ^—, 


The  direction  a  of  the  resultant  force  may  be  found  from  this 
equation.     Thus  R  is  completely  determined. 
When  6  =  90°,  equation  (1)  reduces  to 


(4)  R  =      fi*  +  Ft. 
We  also  have 

F2  Fl 

(5)  sin  a  =  —  ,  cos  a  =  —  . 

ti  T 

Two  forces  which  have  a  given  force  for  their  resultant  are 
called  the  components  of  this  force.  Thus  FI  and  F2  are  com- 
ponents of  R.  The  process  of  finding  the  resultant  of  any 
number  of  forces  is  known  as  the  composition  of  forces.  The 
process  of  finding  the  components  of  a  given  force  is  called  the 
resolution  of  forces.  Two  systems  of  forces  acting  on  a  particle 
and  having  the  same  resultant  are  said  to  be  equivalent. 


VII,   §129] 


STATICS 


157 


FIG.  7& 


129.  Rectangular  Components  of  a  Force.     Frequently, 
it  is  desired  to  resolve  a  force 
into  components  which  are,  re- 
spectively, parallel  and  perpen- 
dicular to  a  given  line.     Such 
components  are    called   rectan- 
gular components.     In  this  case     — 
the  magnitudes  of  the  compon- 
ents may  be  found  by  the  solu- 
tion of  equations  (5),  or  directly  from  a  figure.     See  Fig.  78. 
Thus  we  find 
(6)  Fi  =  R  cos  a,  F2  =  R  sin  a. 

These  formulas  give  FI  and  F2  as  the  rectangular  components 
of  R. 

Similarly  the  component  of  any  given  force  along  any  given 
line  is  equal  to  the  magnitude  of  the  force  multiplied  by  the 
cosine  of  the  angle  between  the  line  and  the  force. 

EXERCISES 

1.  Given  F!  =  48.7,  F2  =  69.8,  6  =  65°  20  ,  find  R  and  a. 

2.  Given  FI  =  20.3,  F2  =  60.2,  0  =  135°  10';  find  R  and  a. 

3.  Given  FI  =  60.3,  F2  =  30.2,  0  =90°,  find  R  and  «. 

4.  Given  F!  =  26.7,  F2  =  45  7,  0  =  60°;  find  R  and  a. 

5.  R  =  140,  a  =  15°;  find  Fi  and  Fi. 

Ans.  Fi  =  135.2,  F,  =  36.2 

6.  R  =  125,  a  =  25°;  find  Fi  and  FI. 

Ans.  Fi  =  113.3,  F,  =  52.8 

7.  R  =  325,  a  =  35°;  find  Fi  and  F,. 

Ans.  F!  =  266.2,  F,  =  186  4 

8.  R  =  600,  a  =  55°;  find  Fi  and  F2. 

Ans.  F,  =  344.1,  F,  =  491.5 

9.  A  particle  is  acted  upon  by  two  forces,  of  8  and  10  pounds  re- 
spectively, making  an  angle  of  30°  with  each  other.     Find  the  mag- 
nitude of  the  resultant.  Ana.  17.39 


158  MATHEMATICS  [VII,  §  129 

10.  A  boat  is  being  towed  by  two  ropes  making  an  angle  of  60°  with 
each  other.     The  pull  on  one  rope  is  SCO  pounds,  the  pull  on  the  other 
is  300  pounds.     In  what  direction  will  the  boat  tend  to  move?     What 
single  force  would  produce  the  same  result?  [MILLER-LILLY] 

Ans.  21°  47'  with  force  of  500  Ibs.;  700  Ibs. 

11.  Let  a  raft  move  in  a  straight  line  down  stream  with  a  uniform 
speed  of  2  feet  per  second;  suppose  a  man  upon  the  raft  walks  at  a 
uniform  speed  of  4  feet  per  second  in  a  direction  making  an  angle  of  60° 
with  the  direction  of  movement  of  the  raft.     Find  the  speed  and 
direction  qf_the  man  relative  to  the  earth. 

Ans.   V28  ft.  per  sec.  at  an  angle  of  40°  54'  with  direction  of  raft. 

12.  A  river  one  mile  wide  flows  at  a  rate  of  2.3  miles  per  hour.     A 
man,  who  in  still  water  can  row  4.2  miles  per  hour,  desires  to  cross  to  a 
point  directly  opposite.     Find  in  what  direction  he  must  row  and  how 
long  he  will  be  in  crossing. 

Ans.  Upstream  at  an  angle  of  56°  48'  with  direction  of  stream;  17 
minutes  approximately. 

13.  A  man  in  a  house  observes  rain  drops  falling  with  a  speed  of 
32  feet  per  second.     The  direction  of  descent  makes  an  angle  of  30° 
with  the  vertical.     Find  the  velocity  of  the  wind. 

Ans.  18.5  ft.  per  sec. 

14.  A  motor  boat  points  directly  across  a  river  which  flows  at  the 
rate  of  3.5  miles  per  hour;  the  boat  has  a  speed  in  still  water  of  10  miles 
per  hour.     Find  the  speed  of  the  boat  and  the  direction  of  its  motion. 

Ans.  10.59  mi.  per  hr.,  70°  43'  with  direction  of  stream. 

15.  From  a  railway  train  going  40  mi.  per  hour  a  bullet  is  fired 
2,000  ft.  per  second  at  an  angle  of  65°  with  the  track  ahead.     Find  its 
speed  and  direction. 

130.  Triangle  of  Forces.     It  will  be  seen  at  once  on  re- 
ferring to   Fig.  77  that  the  sum  or 
resultant  of  the  two  forces  FI  and  F2 
could    be    obtained    more    easily    by 
7g  drawing  a  triangle  ABD,  as  in  Fig.  79; 

when  applied  to  find  the  resultant  of 

two  forces  the  triangle  ABD  is  called  the  triangle  of  forces. 
Referring  again  to  Fig.  79,  it  is  evident  that  if  a  force  equal 


VII,  §130]  STATICS  159 

and  opposite  to  the  resultant  R  were  applied  at  A,  this  force 
and  the  forces  FI  and  Fz  would  balance,  and  the  point  A  would 
be  in  equilibrium.  Another  way  of  stating  the  proposition 
would  be  as  follows. 

//  three  concurrent  forces  are  in  equilibrium,  their  magnitudes 
arc  proportional  to  the  three  sides  of  a  triangle  whose  sides,  taken 
in  order,  are  parallel  to  the  directions  of  the  given  forces.  Con- 
rcrwly,  if  the  magnitudes  of  three  concurrent  forces  are  propor- 
tional to  the  three  sides  of  a  triangle  and  their  directions  are  paral- 
lel to  the  sides  taken  in  order,  these  forces  will  be  in  equilibrium. 

EXERCISES 

1.  Draw  a  triangle  ABC  whose  sides  BC,  CA,  AB  are  7,  9,  11  units 
long.     If  ABC  is  a  triangle  for  three  forces  in  equilibrium  at  a  point  P, 
and  if  the  force  corresponding  to  the  side  BC  is  a  force  of  21  Ibs.,  show 
in  a  diagram  how  the  forces  act,  and  find  the  magnitude  of  the  other 
two  forces.  Ans.  27,  33. 

2.  Draw  two  lines  AB  and  AC  containing  an  angle  of  120°,  and  sup- 
pose a  force  of  7  units  to  act  from  A  to  B  and  a  force  of  10  units  from 
A  to  C.     Find  by  construction  the  resultant  of  the  forces,  and  the 
number  of  degrees  in  the  angle  its  direction  makes  with  AB. 

Aris.   V79;  77°,  approximately. 

3.  Draw  an  equilateral  triangle  ABC,  and  produce  BC  to  D,  making 
CD  equal  to  BC.     Suppose  that  BD  is  a  rod  (without  weight)  kept  at 
rest  by  forces  acting  along  the  lines  AB,  AC,  AD.     Given  that  the 
force  acting  at  B  is  one  of  10  units  acting  from  A  to  B,  find  by  con- 
struction (or  otherwise)  the  other  two  forces,  and  specify  them  com- 
pletely. 

4.  Find  the  resultant  of  two  velocities  of  9  and  7  ft.  per  second 
acting  at  a  point  at  an  angle  of  120°.  Ans.   ^G7. 

5.  Find  the  magnitude  and  direction  of  the  resultant  of  two  velocities 
of  5  and  4  ft.  per  second  acting  at  a  point  at  an  angle  of  45°. 

Ans.  8.32;  19°  52'. 

6.  A  certain  clothes  line  which  is  capable  of  withstanding  a  pull  of 
300  pounds,  is  attached  to  the  ends  A  and  B  of  two  posts  40  feet  apart, 


160 


MATHEMATICS 


[VII,  §  130 


A  and  B  being  in  the  same  horizontal  line.  When  the  rope  is  held 
taut  by  a  weight  W,  attached  to  the  middle  point,  C,  of  the  line,  C  is 
four  feet  below  the  horizontal  line  AB.  Find  the  weight  of  the  heaviest 
boy  it  will  support  without  breaking.  [MILLER-LILLY] 

Ans.    117.7,  Ibs. 

7.  A  street  lamp  weighing  100  pounds  is  supported  by  means  of  a 
pulley  which  runs  smoothly  on  a  cable  supported  at  A  and  B,  on  oppo- 
site sides  of  the  street.     If  A  is  10  feet  above  B,  and  the  street  60  feet 
wide,  and  the  cable  75  feet  long,  find  the  point  on  the  cable  where  the 
pulley  rests,  and  the  tension  in  the  cable.  [MILLER-LILLY] 

8.  A  particle  of  weight  W  lies  on  a  smooth  plane  which  makes  an 
angle  a  with  the  horizon.     Show  that  P  =  W  sin  a,  R  —  W  cos  a, 
where  P  is  the  force  acting  along  the  plane  to  keep  the  particle  from 
slipping  and  R  is  the  reaction  of  the  plane. 

131.  The  Simple  Crane.  One  of  the  most  useful  applica- 
tions of  the  triangle  of  forces  is  the  case  of  an  ordinary  crane. 
It  has  a  fixed  upright  member  AB  called  the  crane  post,  a  member 
AC  called  the  jib,  and  a  tie-rod  BC,  A  weight  W  suspended 


FIG.  80 

rigidly  at  C  is  kept  in  position  by  three  forces  in  equilibrium. 
These  forces  are  (a)  the  weight  W,  (b)  the  pull  in  the  tie-rod, 
and  (c)  the  thrust  in  the  jib.  To  determine  their  magnitudes 
construct  to  scale  a  force  triangle  EFG.  Draw  EF  parallel  to 
the  line  of  action  of  the  weight  W  and  equal  to  W  in  magni- 
tude. From  F  draw  F G  parallel  to  the  jib  and  from  E  draw 


VII,  §  131]  STATICS  161 

EG  parallel  to  the  tie-rod.  The  lengths  of  EG  and  FG  to  the 
same  scale  on  which  EF  was  drawn  represent  the  thrust  in  the 
jib  and  the  pull  in  the  tie-rod.  The  directions  of  the  forces 
acting  along  the  tie-rod  and  jib  are  given  by  following  around 
the  triangle  in  order  from  E  to  F  to  G  to  E. 

When  a  crane  is  used  to  raise  or  lower  a  weight,  the  weight 
is  held  by  a  rope  passing  over  a  pulley  at  C.  The  tension  of  the 
rope  must  now  be  taken  into  account. 

Suppose  a  chain  or  rope  supporting  the  weight  is  made  to 
pass  over  a  pulley  at  C,  and  is  then  led  on  to  a  drum  at  A  round 
which  the  rope  or  chain  is  coiled.  The  pull  in  the  rope  and 
tie-rod  together  is  the  same  as  before  and  is  represented  by  EG. 
The  tension  in  the  rope  is  the  same  on  each  side  of  the  pulley. 
Therefore  if  we  mark  off  on  EG  a  distance  HE  equal  to  EF, 
this  distance  will  represent  the  pull  in  the  rope,  thus  leaving 
GH  to  represent  the  pull  in  the  tie-rod. 

EXERCISES 

Find  the  pull  in  the  tie-rod  and  the  thrust  in  the  jib  of  a  crane  when 
the  dimensions  and  weight  are  as  given  below.  (Weight  suspended 
rigidly  at  C.) 

1.  AB  =  10,     BC  =  24,     AC  =  31,     W  =  12  tons. 

2.  AB  =6,     BC  =  12,     AC  =  16,     W  =  6  tons. 

3.  AB  =  15,     BC  =  50,     AC  =  45,     W  =  5  tons. 

4.  AB  =    9,     BC  =  16,     AC  =  21,     W  =  4  tons. 

5.  The  jib  of  a  crane  is  subjected  to  a  compressive  force  equal  to 
the  weight  of  24  tons,  the  suspended  load  being  10  tons.     If  the  in- 
clination of  the  jib  to  the  horizontal  is  60°,  find  the  tension  in  the  tie- 
rod.  Ans.  16.1  tons. 

6.  In  a  crane  the  pull  in  the  tie-rod  inclined  at  an  angle  of  60°  to 
the  vertical  is  18  tons.     If  the  weight  lifted  be  8  tons,  find  the  thrust 
in  the  jib.  Ans.   23.06  tons. 

7.  In  exercises  1-4,  find  the  forces  acting  in  each  member  of  the 
crane  when  the  load  is  suspended,  but  not  rigidly,  at  the  jib  head,  for 

12 


162 


MATHEMATICS 


[VII,  §  131 


the  two  cases  when  the  rope  passes  from  the  pulleys  to  the  drum  (a)  par- 
allel to  the  tie-rod,  (6)  parallel  to  the  jib. 

8.  The  jib  of  a  crane  is  subjected  to  a  compressive  force  equal  to  the 
weight  of  4000  Ibs.,  the  suspended  load  being  2000  Ibs.      If  the  inclina- 
tion of  the  jib  to  the  horizontal  is  45°,  find  the  tension  in  the  tie-rod. 

9.  In  a  crane  the  pull  in  the  tie-rod  inclined  45°  to  the  vertical  is 
1000  Ibs.     Find  the  thrust  in  the  jib  if  the  weight  is  2000  Ibs. 

10.  In  Ex.  9  find  the  thrust  in  the  jib  if  the  weight  is  1000  Ibs. 

11.  The  thrust  in  the  jib  inclined  60°  to  the  vertical  is  1800  Ibs. 
The  load  is  900  Ibs.    Find  the  tension  in  the  tie-rod. 

132.  Polygon  of  Forces.  The  resultant  of  three  or  more 
concurrent  forces  lying  in  the  same  plane  may  be  found  by 
repeated  applications  of  the  triangle  of  forces. 

Let  a  particle  at  0  be  acted  upon  by  any  number  of  forces, 
Fi,  F2,  •  •  • ;  to  be  definite,  say  Fi,  F2,  F3,  F4.  To  find  their 
resultant  proceed  as  follows.  For  the  forces  FI  and  F2  con- 
struct the  triangle  of  forces  OAB  (Fig.  81).  Then  OB  is  the 


FIG.  81 

resultant  of  FI  and  F2.  For  the  forces  OB  and  F3  construct 
the  triangle  of  forces  OBC.  The  sum  is  given  by  OC.  In  a 
similar  manner  combine  OC  and  F4.  The  resultant  is  R  =  OD. 
The  construction  of  the  lines  OB  and  OC  is  unnecessary  and 
should  be  omitted.  The  figure  OABCDO  is  called  the  polygon 
of  forces.  OD,  the  closing  side,  is  called  the  resultant.  It  will 
be  noticed  that  the  arrows  on  the  vectors  representing  the 


VII,  §133] 


STATICS 


163 


given  forces  all  run  in  the  same  sense  around  the  polygon, 
while  the  arrow  of  the  resultant  runs  in  the  opposite  sense. 

If  any  number  of  forces  acting  at  a  point  can  be  represented 
by  the  sides  of  a  closed  polygon  taken  in  order,  the  point  is  in 
equilibrium  and  the  resultant  is  zero. 

From  the  above  discussion  we  obtain  the  following  rule  for 
finding  the  resultant  of  any  number  of  forces. 

From  any  point  0  draw  a  line  OA  to  represent  in  magnitude 
and  direction  the  force  F\.  From  the  extremity  A  draw  a  linc~  AB 
to  represent  in  magnitude  and  direction  the  force  F%.  Continue 
thix  process  for  each  of  the  given  system  of  forces.  Then  the  line 
which  it  is  necessary  to  draw  from  0  to  close  the  polygon  represent* 
the  resultant  in  magnitude  and  direction. 

133.  Resultant  of  Several  Concurrent  Forces.  Analytic 
Formula.  Let  any  number  of  forces  Fi,  F2,  •••,  lying  in  the 
same  plane,  act  on  a  particle  at  0.  To  fix  the  ideas,  suppose 
there  are  three  forces.  With  0  as  origin  refer  the  forces  to  a 
pair  of  coordinate  axes,  OX  and  OY  (Fig.  82).  Resolve  each 


force  into  two  components,  one  along  OX  and  one  along  OY. 
The  components  of  F!  will  be  OA  and  OB;  of  Fz,  OC  and  OE; 


164  MATHEMATICS  [VII,  §  133 

of  FS,  OD  and  OF.     If  a\,  a^,  a3  represents  the  angles  which 
FI,  FZ,  FS  make  respectively  with  the  axis  0  X,  we  have : 

Xt  =  OA  =  FI  cos  ai,  F!  =  OB  =  Fl  sin  ah 

Xz  =  OE  =  Fz  cos  «2,  F2  =  OC  =  Fz  sin  a2, 

Z3  =  0Z>  =  F3  cos  as,  Y3  =  OF  =  F3  sin  a3. 

If  a  component  acts  upward  or  toward  the  right  we  will  assume 
it  to  be  positive ;  if  downward  or  toward  the  left,  negative. 

The  given  system  of  forces  is  equivalent  to  another  set  con- 
sisting of  the  rectangular  components  of  the  forces  of  the  given 
system.  Let  us  use  the  letters  X  and  Y  to  represent  the  sum 
of  these  components  along  the  x-axis  and  the  7/-axis,  respectively. 
Then 


(7) 


X   =   FI  COS  «i   +  Fz  COS  «2   +  FZ  COS  «3 

=  the  sum  of  all  the  horizontal  components. 

Y  =  FI  sin  «i  +  FZ  sin  «2  +  F3  sin  a3 

=  the  sum  of  all  the  vertical  components 


The  two  forces  X  and  Y  acting  at  right  angles  to  each  other 
are  equivalent  to  the  given  system  of  forces.  The  single  force 
R  which  is  the  resultant  of  X  and  Y  is  also  the  resultant  of  the 
given  system  of  forces.  We  have 


(8)  R  =       X2  +   P. 

The  resultant  R  is  always  thought  of  as  being  positive.  We 
now  have  the  magnitude  of  the  resultant  force.  To  find  the 
line  of  action  we  have 

Y 

(9)  tana  =  — , 

Ji. 

where  a  is  the  angle  between  the  positive  direction  of  the  x-axis 
and  the  positive  direction  of  the  resultant  R. 


VII,  §  134]  STATICS  165 

To  find  the  direction  along  the  line  of  action  the  two  following 
equations  are  used : 

Y  IT 

(10)  sin  a  =  — ,         cos  a  =  — . 

It  is  obvious  that  equations  (10)  determine  both  the  line  of 
action  and  the  direction  along  that  line. 

EXERCISES 

1.  If  four  forces  of  5,  6,  8,  and  11  units  make  angles  of  30°,  120°, 
225°,  and  300°  respectively,  with  a  fixed  horizontal  line,  find  the  mag- 
nitude and  the  direction  of  the  resultant.  Ans.   7.39 ;    —  81°  6'. 

2.  Forces  P,  2P,  3P,  and  4P  act  along  the  sides  of  a  square  taken 
in  order.     Find  the  magnitude,  the  direction,  and  the  line  of  action  of 
the  resultant. 

Ana.  2V2P,  -  45°  with  line  of  force  of  4P,  through  (-  2a,  -  4a) 
where  side  of  square  is  4o  and  origin  of  coordinates  is  intersection  of 
3P,  4P. 

3.  A  particle  is  acted  on  by  five  coplanar  forces ;  a  force  of  5  Ibs. 
acting  horizontally  to  the  right,  and  forces  of  1,  2,  3,  4  Ibs.  making 
angles  of  45°,  60°,  225°,  and  300°  respectively  with  the  5-lb.  force. 
Find  the  magnitude  and  the  direction  of  the  resultant. 

Ans.   R  =  7.31,  0  =  334°  28'. 

4.  Find  the  resultant  of  the  following  concurrent,  coplanar  forces : 
(a)   (14,  45°),  (6,  120°),  (5,  240°). 

(6)   (2,  0°),  (3,  50°),  (4,  150°),  (5,  240°). 

(c)  (2,  -  30°),  (3,  90°),  (4,  135°),  (5,  225°). 

(d)  (5,  -  30°),  (6,  270°),  (4,  120°),  (3,  135°). 

134.  Resultant  of  Parallel  Forces.  Let  FI  and  F2  be  two 
parallel  forces  acting  in  the  same  direction  and  with  their 
points  of  application  at  the  points  A  and  B,  Fig.  83.  At  A 
and  B  apply  two  equal  and  opposite  forces,  AS  and  BT,  whose 
line  of  action  coincides  with  AB.  These  will  balance  and  will 
not  change  the  effect  of  the  other  forces.  Find  the  resultant 
AD  of  AS  and  FI,  and  the  resultant  BE  of  B  T  and  F2,  by  con- 
structing the  parallelograms  of  forces.  Then  by  constructing  a 


166 


MATHEMATICS 


[VII,  §134 


parallelogram  of  forces  at  0,  the  intersection  of  AD  and  BE 
produced,  we  may  find  their  resultant  OR,  which  is  evidently 
the  resultant  of  FI  and  F2.  Draw  MK  parallel  to  AB.  Then 

E 


since  OM  is  equal  to  AD  in  magnitude  and  in  direction  and  MR 
is  equal  to  BE  in  magnitude  and  direction,  it  follows  that  the 
triangles  OMK  and  ADFi  are  equal,  and  the  triangles  MKR 
and  BTE  are  equal.  Hence  the  resultant  OR  is  equal  to 
FI  +  F2,  and  its  line  of  action  is  a  line  through  the  point  0 
parallel  to  the  lines  of  action  of  FI  and  F2. 

Let  C  be  the  intersection  of  AB  and  OR.  Then  from  the 
pairs  of  similar  triangles  OCA  and  AFiD,  and  OCB  and  BF2E, 
we  have 

AC      AS  BC  _BT  _  AS 

oc~J\  oc  "  FT  ~  77 ' 

Hence 

Fi      BC 

(ii)  F  =  IF" 

r  2        ^10 

A  similar  proof  can  be  given  for  the  case  of  unequal  parallel 
forces  acting  in  opposite  directions.  Both  results  may  be 
combined  into  the  following  theorem. 

The  resultant  of  any  two  parallel  forces,  acting  in  the  same 
direction,  or  of  two  unequal  forces  acting  in  opposite  directions, 


VII,  §136]  STATICS  167 

is  parallel  to  the  forces  and  equal  to  their  algebraic  sum  and  cuts 
a  line  joining  their  points  of  application  into  segments,  the  lengths 
of  which  are  inversely  proportional  to  the  magnitudes  of  the  forces. 

135.  Moment  of  a  Force.     The  moment  of  a  force  with  re- 
spect to  a  point,  called  the  center  of  moments,  is  the  product  of 
the  magnitude  of  the  force  and  the  perpendicular  distance,  called 
the  arm,  from  the  point  to  the  line  of  action  of  the  force. 

Geometrically  the  moment  of  a  force  is  represented  by  twice 
the  area  of  a  triangle  whose  base  is  the  line  representing  the  given 
force  and  whose  vertex  is  the  center  of  moments. 

The  moment  of  a  force  in  a  given  plane  with  respect  to  a  line 
perpendicular  to  that  plane  is  the  moment  of  the  force  with 
respect  to  the  foot  of  that  perpendicular.  The  line  is  called  the 
axis  of  moments. 

Moments  are  positive  or  negative  according  as  they  tend  to 
produce  counter  clockwise  or  clockwise  rotation  about  the  axis 
of  moments. 

136.  Composition  of  Moments.     The  algebraic  sum  of  the 
moments  of  any  two  forces  with  respect  to  any  point  of  their  plane 
is  equal  to  the  moment  of  their  resultant  with  respect  to  the  same 
point. 

There  are  two  cases. 

CASE  1.  When  the  lines  of  action  of 
the  forces  are  not  parallel. 

PROOF.  Let  OP,  OQ  be  two  forces 
acting  at  0,  and  OR  their  resultant;  and 
let  A  be  any  point  in  the  plane  about 

which  moments  are  to  be  taken.  Join  AO,  AP,  AQ,  and  AR. 
Then 

Area  &AOQ  =  Area  &APR  +  Area  ARPO,* 

*By  convention  areas  are  positive  or  negative  according  as  their  boundaries  are 
travel sed  in  counterclockwise  or  clockwise  direction. 


168  MATHEMATICS  [VII,  §  136 

since  they  have  equal  bases  OQ  and  PR,  and  the  altitude  of 
AAOQ  is  equal  to  the  sum  of  the  altitudes  of  APR  and  RPO. 

Area  AAOR  =  Area  AAOP  +  Area  &APR  +  Area  ARPO, 
for  obvious  reasons  it  follows  that 

Area  AAOR  =  Area  LAOP  +  Area  AAOQ. 

Therefore  the  moment  of  OR  about  A  is  equal  to  the  sum 
of  the  moments  of  OP  and  OQ  about  A. 

Frequently  it  is  easier  to  determine  the  moment  of  a  force 
by  computing  the  sum  of  the  moments  of  its  components  than 
to  determine  it  directly. 

CASE  II.  When  the  lines  of  action  of  the  forces  are  parallel. 
We  exclude  the  case  in  which  the  forces  are  equal  and  opposite. 

Suppose  that  two  forces  P  and  Q  act  on  the  body  at  the 
points  A  and  B,  Fig.  85.  From  any  point  0,  draw  OACB  per- 
pendicular to  the  lines  of  action  of  the  forces.  Let  OA  =  p, 
AC  =  x.  Then  by  §  134,  CB  =  Px/Q.  Taking  moments  about 
0  we  find 

moment  of  P  =  P  •  p,          moment  of  Q  =  Q(p  +  x  + 
moment  of  P  +  moment  of  Q  =  Pp  -\-Qp-\-Qx-\-Px 


=  moment  of  R. 

If  P  and  Q  are  in  opposite  directions  the  proof  is  similar  to  the 
above  and  is  left  to  the  student.  The  proof  in  case  P  and  Q 
are  equal  but  opposite  in  direction  is  given  in  the  following 
section. 


f 

It 

r 

- 

FIG.  86 

6 

A 

FIG.  85 

0      B 

137.   Couples.     A  system  of  two  parallel  forces,  equal  irt 


VII,  §138]  STATICS  169 

magnitude  and  opposite  in  direction,  is  called  a  couple.  The 
perpendicular  distance  between  the  lines  of  action  of  the  forces 
is  called  the  arm  of  the  couple;  and  the  plane  containing  the 
forces  is  called  the  plane  of  the  couple. 

The  moment  of  a  couple  is  the  algebraic  sum  of  the  moments 
of  its  forces  about  any  axis  perpendicular  to  its  plane  and  is 
equal  to  the  product  of  either  force  and  the  length  of  the  arm.  For, 
let  0  be  any  axis,  perpendicular  to  the  plane  of  the  couple,  and 
OA  and  OB,  the  moment  arms  of  the  forces  with  respect  to  0. 
Taking  moments  about  0,  we  have 

F-OB  -  F-'OA  =  F-AB. 

The  sign  of  the  couple  is  plus  if  it  tends  to  turn  with  clock- 
wise rotation,  and  minus  if  it  tends  to  turn  with  counter-clock- 
wise rotation. 

138.  Conditions  of  Equilibrium. 

(a)  Concurrent  coplanar  forces.  In  order  that  the  forces  of 
a  system  may  balance  each  other,  the  resultant  must  be  equal 
to  zero,  that  is 


(12)  R  =  V(SZ)2  +  (S7)2  =  0. 
Hence  we  have  also 

(13)  SX  =  0,         and         2Y  =  0. 

The  algebraic  sum  of  the  moments  of  the  forces  (written  SAf) 
about  any  point  is  equal  to  the  moment  of  the  resultant.  If 
the  forces  are  in  equilibrium,  R  =  0;  therefore 

(14)  SM  =  0. 

These  conditions  are  used  in  the  second  method  of  Ex.  1,  below. 

(6)   System  of  parallel  forces.     If  the  algebraic  sum  of  a  sys- 

tem of  parallel  forces  is  not  zero,  the  resultant  is  a  single  force 

and  the  system  is  not  in  equilibrium.     Hence  a  necessary  con- 


170 


MATHEMATICS 


[VII,  §  138 


dition  for  equilibrium  is  that 


=  0, 


where  F  represents  the  magnitude  of  a  force.  If  the  algebraic 
sum  of  the  moments  of  the  forces  about  any  point  is  not  zero, 
while  the  algebraic  sum  of  the  forces  is  zero,  the  resultant  is  a 
couple,  and  the  body  is  not  in  equilibrium.  Hence  a  necessary 
condition  for  equilibrium  is  that 
(15)  2F  •  x  =  0, 

where  x  is  the  moment  arm  of  the  force  F. 

EXERCISES 

BALANCED  SYSTEMS  OF  FORCES  ACTING  THROUGH  THE  SAME  POINT 

1.   A  triangular  frame  ABC  (Fig.  87)  carries  a  load  of  1000  Ibs.  at  A. 
Find  the  stresses  in  the  members  AB  and  AC. 


1000 


FIG.  87 


FIG.  88 


SOLUTION.  We  have  in  this  problem  a  balanced  system  of  forces 
acting  through  the  point  A,  namely,  the  load  of  1000  Ibs.  and  the  forces 
FI  and  F2  in  the  members  AC  and  AB.  Both  AC  and  AB  are  subjected 
to  a  compression.  Hence  both  members  exert  a  thrust  in  the  direction 
indictated  by  the  arrows.  The  problem  is  to  determine  the  magnitude 
of  two  unknown  forces  in  a  balanced  system  of  three  forces,  the  direc- 
tions of  the  forces  being  known.  This  problem  may  be  solved  in  any 
one  of  the  three  following  ways. 

FIRST  METHOD.     (Triangle  of  Forces.)     The  forces  may  be  repre- 


VII,  §  138] 


STATICS 


171 


sented  by  the  sides  of  a  triangle  taken  in  order,  Fig.  88.  If  the  figure 
is  drawn  to  scale  the  magnitudes  of  the  unknown  forces  F\  and  Ft  may 
be  obtained  directly  from  the  figure  by  measurement. 


woo 


FIG.  89 


1000 
FIG.  90 


If  the  lengths  of  all  of  the  members  of  the  frame  ABC  are  known  or 
can  be  computed,  we  can  obtain  the  magnitudes  of  FI  and  Ft  by  pro- 
portion, since  the  triangle  ABC  and  the  force  triangle  are  similar. 

In  this  particular  problem  we  observe  that  the  force  triangle  is  right- 
angled  and  one  acute  angle  is  60°.  Hence 

F!  =  1000  sin  60°  =  866  Ibs.,     F2  =  1000  cos  60°  =  500  Ibs. 

SECOND  METHOD.  (Resolution  of  Forces.)  Refer  the  forces  to  a 
system  of  coordinate  axes,  Fig.  88,  and  use  the  conditions  (13)  of  equi- 
librium. We  have 

SX  =  F2  cos  30°  -  F!  cos  60°  =  0, 

2F  =  Ft  sin  30°  +  Fl  sin  60°  -  1000=  0. 

The  solution  of  these  equations  gives, 

Fl  =  866  Ibs.,  Ft  =  500  Ibs. 

THIRD  METHOD.  (Moments.)  The  sum  of  the  moments  of  all  the 
forces  about  any  arbitrarily  chosen  point  leads  to  one  equation  contain- 
ing the  unknowns.  If  we  take  the  sum  of  the  moments  of  all  the  forces 
about  as  many  arbitrary  points  as  there  are  unknowns  then  we  will  have 
as  many  equations  as  unknowns.  The  solution  of  these  equations  gives 
the  magnitudes  of  the  unknown  forces.  It  is  often  advantageous  to 
choose  for  the  points  about  which  moments  are  taken,  points  on  the  lines 
of  action  of  the  unknown  forces,  one  on  each  line. 

Taking  moments  about  B  we  find 


172 


MATHEMATICS 


[VII,  §  138 


whence 


SAf  =  8V3F,  -  1000  X  12  =  0, 
Fi  =  866  Ibs. 


Taking  moments  about  C  we  find 

SM  =  1000  X  4  -  8F2  =  0, 


whence 


F2  =  500  Ibs. 


2.    Find  the  stresses  in  the  members  AB  and  AC,  of  the  triangular 

frame  ABC,  Fig.  91,  the  load  at  A  being 
1000  Ibs. 

[HiNT.     Use  the  triangle  of  forces.] 
Ans.   AB,  739.1  Ibs.;  AC,  922.2 

3.  S  Ive    Ex.    2    (a)    by    using    the 
method  of  resoluti  n  of  forces ;    (6)  by 
the  method  of  moments. 

4.  Assuming  that  the  frame  in  Ex. 
2  is  supported  by  a  vertical  force  at  B,  find  the  magnitude  of  the  force 
and  the  stress  in  BC. 

5.  A  crane  is  loaded  with  3000  Ibs.  at  C.     Determine  the  stresses 
in  the  boom  CD,  the  tie  BC,  the  mast  BD 

and  the  stay  AB,  Fig.  92. 

[HiNT.     Use  the  triangle  of  forces.] 

Ans.  CD,  6250  Ibs.  (compression) ;  BC, 
4250  Ibs.  (tension) ;  AB,  5858  Ibs.  (tension) ; 
BD,  2500  Ibs.  (compression). 

6.  Solve  Ex.  5,  using  the  method  of  reso- 
lution of  forces. 

7.  Find  the  horizontal  and  vertical  components  of  the  supporting 

forces  at  A  and  D,  Ex.  5. 

8.  Find  the  stresses  in  the  members  of  the 
crane  in  Ex.  5,  when  the  boom  makes  an  an- 
gle of  15°  with  the  horizontal. 

9.    What  is  the  smallest  force  F  which  will 
prevent  a  weight  of  150  Ibs.  from  slipping 

down  the  incline  represented  in  Fig.  93  if  friction  is  neglected? 

Ans.   212.2  Ibs. 


FIG. 


VII,  §  138]  STATICS  173 

10.  Let  F  =  150  Ibs.  (Fig.  93)  and  let  the  weight  also  be  150  Ibs 
What  will  be  the  largest  angle  between  the  inclined  plane  and  the  hori 
zontal  at  which  the  weight  will  not  slip  ?  Ans.   30°. 

11.  Experiments  indicate  that  a  horse  exerts  a  pull  on  his  traces 
equal  to  about  one-tenth  of  his  weight,  when  the  working  day  does  not 
exceed  10  hours.     The  draft  of  a  certain  wagon  is  due  to  (a)  axle 
friction  =  5  Ibs.  per  2000  Ib.  load ;  (6)  gradient  or  hills ;  (c)  rolling  draft 
depending  on  height  of  wheel,  width  of  tire,  condition  of  road-bed,  etc. 

How  large  a  load  can  a  team  of  horses  each  weighing  1000  Ibs.  pull 
up  a  10%  grade  if  the  rolling  draft  is  zero.  (A  10%  grade  is  a  rise  of 
10  feet  for  each  100  feet  measured  horizontally  along  the  roadway.) 

Ans.  1961  Ibs. 


FIG.  94 

12.  What  extra  pull  must  a  horse  exert  on  his  traces  (assumed 
horizontal)  if  on  a  level  road  the  wheel,  4  feet  in  diameter,  strikes  a 
stone  2  inches  high,  the  load  being  1000  Ibs.  Ans.   436  Ibs. 

13.  A  carriage  wheel  whose  weight  is  W  and  whose  radius  is  r  rests 
on  a  level  road.     Show  that  any  horizontal  force  acting  through  the 
center  of  the  wheel  greater  than 


r  —  h 

will  pull  it  over  an  obstacle  whose  height  is  h. 

14.  In  Ex.  13,  let  P  =  100  Ibs.,  W  =  1000  Ibs.,  r  =  2  feet.     Find  h. 

Ans.   0.126  in. 

15.  A  50  Ib.  boy  swings  on  the  middle  of  a  clothes  line  which  is  50  feet 
long.     The  lowest  point  is  2  feet  below  either  end.     Find  the  tension 
in  the  rope.  Ana.   625  Ibs. 

16.  A  wire  90  feet  long  carries  a  weight  of  25  Ibs.  at  each  of  its  trisec- 
tion  points.     When  the  wire  is  taut  each  weight  is  5  feet  below  the  hori- 
zontal line  connecting  the  points  of  support.     Find  the  tension  in  each 
segment  of  the  wire.  Ans.   150 ;  147.9 ;  150  Ibs. 


174 


MATHEMATICS 


[VII,  §138 


17.  Steam  in  the  cylinder  of  an  engine  exerts  a  pressure  of  20,000 
pounds  on  the  piston-head.     The  guides  N,  Fig.  95,  are  smooth.     What 


N 

1 

;  i 

N 
FIG.  95 

is  the  thrust  in  the  connecting  rod  when  it  makes  an  angle  of  20°  with 
the  horizontal?  What  is  the  pressure  on  the  guides  N  ?   [MILLER-LILLY] 

PARALLEL  FORCES  ACTING  IN  THE  SAME  PLANE 
18.  Determine  the  resultant  R  of  each  of  the  following    systems  of 
parallel  forces. 

50  20  30 


Y~—4—  -»!<—  - 


6'—  ->f -5-'  — 


20 


40 


80 


(a)  FIG.  96 

10 


60 

(6)  FIG.  97 


70 


500 
U-3./4- +•__  ... „.• 

800 

(c)  FIG.  98 


19.  Let  AB  (Fig.  99)  represent  a  beam  carrying  the  weights  indi- 
cated and  supported  by  the  vertical  forces  FI  and  F2.     Find  FI  and  F2. 


1000         2000 
>*-»* — 4+ — 4* 


F0=  2500 


FIG.  99 


20.  The  system  of  parallel  forces  in  Fig.  100  is  in  equilibrium.    Find 
the  magnitudes  and  directions  of  the  unknown  forces  FI  and  Ft. 
4P  F, 


FIG.  100 


VII,  §  138] 


STATICS 


175 


21.  If  a  horse  exerts  a  pull  on  his  traces  equal  to  one-tenth  of  his 
weight,  where  should  the  single-tree  for  each  of  two  horses  weighing 
1200  and  1600  Ibs.,  respectively,  be  fastened  to  a  double-tree  in  order 
that  each  horse  shall  do  his  proper  share  of  the  work  ? 

22.  The  center  clevis  pin  A,  of  a  double-tree  is  a  inches  in  front  of  the 
mid-point  B,  of  the  line  connecting  the  end  clevis  pins  C  and  D,  which 
are  b  inches  apart.     If  one  horse  is  pulling  c  inches  ahead  of  the  other 
what  fraction  of  the  load  L  is  each  horse  pulling,  Fig.  101  ? 

1  _         ac 
2 


Ans. 

2 


-  c2 


O 


FIG.  101 

23.  Find  what  fractional  part  of  the  load  each  horse  is  pulling  if 
a  =  2,  when 

(a)  b  =  41,     c  =  9.  (b)  b  =  39,     c  =  15. 

(c)   b  =  34,     c  =  16.  (d)  6  =  52,     c  =  20. 

(e)   b  =  37,     c  =  12.  (/)   b  =  50,     c  =  14. 

(jr)   b  =  61,     c  =  11.  (h)  b  =  36,     c  =  4. 

24.  In  Ex.  22,  if  the  evener  makes  an  angle  0  with  the  tongue,  what 
fractional  part  of  the  load  is  pulled  by  each  horse  ? 


Ans.       + 


\  -  \  tan  9. 


176  MATHEMATICS  [VII,  §  133 

25.  In  Ex.  24  put  a  —  2,  b  —  40.     Plot  a  curve  using  values  of  9  as 
abscissas  and  values  of  the  load  pulled  by  one  horse  as  ordinates.     What 
can  you  say  about  the  part  of  the  load  pulled  by  this  horse  as  0  increases  ? 

26.  In  each  of  the  cases  of  Ex.  23  find  the  pounds  of  pull  exerted  by 
each  horse  if  the  total  pull  on  the  load  is  362.88  Ibs. 

27.  The  middle  clevis  pin  A  of  a  three-horse  evener  is  a  inches  in  front 
of  the  point  B  of  the  line  connecting  the  end  clevis  pins  C  and  D.     The 
end  clevis  pins  are  b  and  26  inches  from  the  point  B.     Find  what  frac- 
tional part  of  the  load  is  borne  by  the  horse  on  the  longer  end  when  it  is 
c  inches  behind  the  other  horses. 


28.  Find  what  fractional  part  of  the  load  the  horse  on  the  long  end  is 
pulling  if  a  =  2,  when 

(a)  b  =  24,     c  =  1,  2,  3,  4,  5,  6,  7,  8,  9,  10. 

(b)  b  =25,     c  =  1,  2,  3,  4,  5,  6,  7,  8,  9,  10. 

(c)  b  =  26,     c  =  2,  4,  6,  8,  10,  12,  14. 

29.  In  Ex.  27,  if  the  evener  makes  an  angle  0  with  the  tongue,  what 
fractional  part  of  the  load  is  pulled  by  the  horse  on  the  long  end. 

Ans.  -  +  --tau6. 
3      3b 

30.  In  Ex.  29  put  a  =  2,  b  =  25.     Plot  a  curve  using  values  of  0 
as  abscissas  and  fractional  parts  of  the  load  pulled  by  the  horse  on  the 
long  end  as  ordinates.     Discuss  the  problem. 

31.  A  steel  rail  60  ft.  long  weighs  1595  Ibs.     Where  must  a  fulcrum 
be  placed  so  that  a  180  Ib.  man  at  one  end  can  raise  4  tons  at  the  other? 

Ans.   6  ft. 


CHAPTER  VIII 
SMALL  ERRORS 

139.  Errors  of  Observation.  Suppose  that  we  measure  the 
length  of  a  building  and  record  the  result.  Such  a  record  is 
called  a  reading  or  an  observation.  Suppose  that  we  measure 
the  same  length  and  record  the  reading  on  each  of  several  suc- 
cessive days.  On  comparison  it  is  likely  we  shall  find  that 
they  do  not  exactly  agree.  What  then  is  the  true  length? 
Whatever  the  actual  length  may  be  the  difference  between  it 
and  any  observation  of  it  is  called  an  error  of  observation. 

Suppose  that  we  measure  the  length  of  a  building  with  a  tape 
whose  smallest  division  is  one  foot.  If  the  length  is  not  a  whole 
number  of  feet,  we  estimate  by  the  eye  the  fraction  of  a  foot  left 
over.  This  estimate  will  almost  certainly  be  in  error.  If  we 
measure  the  same  length  with  a  tape  divided  to  eighths  of  an  inch, 
the  end  of  the  building  may  coincide  with  a  division  of  the  tape 
or  we  may  have  to  estimate  the  fraction  of  an  eighth.  Subse- 
quent readings  are  not  likely  to  agree  exactly  with  the  first,  and 
even  if  they  do  all  agree  we  cannot  be  sure  that  we  have  the  true 
length.  Inattention  and  lack  of  precision  of  the  observer,  in- 
experience in  using  the  measuring  instrument,  or  the  use  of  an 
instrument  which  is  defective  or  out  of  adjustment,  all  tend 
to  introduce  errors.  It  is  important  to  keep  in  mind  that  such 
errors  are  always  present,  in  greater  or  less  degree,  in  every  set 
of  observations. 

If  a  is  the  recorded  reading  of  a  measurement  of  an  unknown 
quantity  u,  a  measure  of  the  error  in  this  reading  is  a  positive 
number  m,  such  that  u  lies  between  a  —  m  and  a  +  m.  The 
actual  error  may  be  very  much  less  than  its  measure  m.  For 
example  if  a  rod  of  (unknown)  length  /  be  measured  with  a  scale 

177 


178  MATHEMATICS  [VIII,  139 

divided  to  tenths  of  an  inch  and  the  reading  is  47.8,  it  is  fairly 
certain  that  47.7  <  /  <  47.9,  and  we  write  I  =  47.8  ±0.1. 

It  is  evident  that  any  number  will  be  in  error  if  it  is  derived  by 
computation  from  other  numbers  which  are  inexact.  Approxi- 
mations are  used  in  computations  not  only  for  recorded  meas- 
urements but  also  in  the  case  of  irrational  numbers,  such  as 
surds,  most  logarithms,  trigonometric  functions,  TT,  etc.  We 
have  3-place,  5-place,  7-place,  10-place  tables  in  order  to  secure 
the  degree  of  accuracy  desired  in  the  computed  result.  In  what 
follows  it  is  shown  how  to  find  a  measure  of  the  error  in  a  number 
computed  by  some  of  the  simpler  processes  of  arithmetic  from 
given  numbers  the  measures  of  whose  errors  are  known. 

140.  Error  in  a  Sum.     Suppose  that  in  measuring  two  quan- 
tities whose  actual  (and  unknown)  values  are  u  and  v,  we  make 
errors  Aw  and  Av  respectively,  and  record  the  readings  a  and  b. 
Then  u  =  a  ±  Aw,  v  =  b  ±  A»  and  their   sum    lies    between 
a  +  b  —  (Aw  +  A0)  and  a  +  b  +  (Aw  +  Au). 

Whence,  w  +  -»  =  a  +  6±  (Aw  +  Aa). 

That  is,  the  error  in  the  sum  of  two  readings  is  measured  by  the 

sum  of  their  errors. 

This  result  is  readily  extended  to  the  sum  of  more  than  two 
readings.  The  error  in  the  difference  of  two  readings  is  never 
greater  than  the  sum  of  their  errors,  though  it  may  be  greater 
than  their  difference. 

EXAMPLE.  Find  the  sum  and  difference  of  46.8  ±  0.65  and  12.4  ± 
0.15.  Here  the  readings  are  4.68,  12.4  and  the  measures  of  their  errors 
are  0.65,  0.15  respectively.  The  measure  of  the  error  of  their  sum  is 

065  +  0.00    =  0.80  ; 

whence          (46.8  ±  0.65)  +  (12.4  ±  0.15)  =  59.2  ±  0.8 
and  (46.8  ±  0.65)  -  (12.4  ±  0.15)  =  34.4  ±  0.8 

141.  Error  in  a  Product.     With  the  same  notation  as  above, 
the  product  uv  lies  between 

ah  —  (aA0  +  feAw  +  Aw  •  At))  and  ab  +  (aAw  +  6Aw  -f  Aw  •  Ac), 


VIII,  §  142]  SMALL  ERRORS  179 

whence,  neglecting  the  small  term  Aw  •  A»,  we  have  approxi- 
mately, 

uv  =  ab  ±  (aAv  +  6  Aw). 

That  is,  a  measure  of  the  error  in  the  product  of  two  readings  is  the 
first  times  the  error  of  the  second  plus  the  second  times  the  error  of 
the  first. 

142.   Error  in  a  Fraction.     The  quotient  of  u  divided  by  v 
evidently  lies  between 

a  —  Au  i  a  +  AM 

6  +  A«  b  —  At)' 

that  is  between 


a      aAfl  +  bAu  j        n   .  aAy  + 


b       b(b  +  At))  6       b(b  -  At>) 

whence  a  measure  of  the  error  in  the  fraction  is 

aAt)  -f  feAw      aAa  -f  &Aw 

—  —  —  -  --  ,  approximately. 

b(b  —  At))  fr2 

That  is,  a  measure  of  the  error  in  the  .quotient  of  two  readings  is  a 
measure  of  their  product  divided  by  the  square  of  the  divisor. 

EXAMPLE.  Find  the  product  and  quotient  of  12.4  ±  0.15  and 
46.8  ±  0.65.  By  §  141,  a  measure  of  the  error  in  the  product  is  (12.4) 
(0.65)  +  (46.  8)  (0.15)  =  15.08  and  the  error  in  their  quotient  is  meas- 
ured by  15.08/(46.8)2  =  0.0069. 

Whence,  (12.4  ±  0.15)  (46.8  ±  0.65)  =  580.32  ±  15.08 

and  (12.4  ±  0.15)/(46.8  ±  0.65)  =  0.265  ±  0.0069 

EXERCISES 

Make  each  of  the  following  computations  and  state  the  result  so  as  to 
show  a  measure  of  the  error  in  it. 

1.    (123  ±  0.2)  ±  (241  ±  0.1).      2.    (222  ±  0.5)  ±  (111  ±  0.4). 
3.    (217  ±  0.2J(117  ±  0.3).  4.    (1267  ±  0.5)(1342  ±  0.4). 

5.    (163  ±  0.2)/(25  ±  0.5).  6.    (732  ±  0.3)/(21  ±  0.4). 

7.  In  Ex.  3  and  4  compute  the  term  Au  •  Ay  neglected. 

8.  In  Ex.  5  and  6  compute  "  ~  A>1  and  a  +  A';\     Find   the   differ- 

0  +  Ay  b  —  Aj 


180 


MATHEMATICS 


[VIII,  §  142 


ence  between  the  error  thus  computed  and  those  computed  in  exercises 
5  and  6  and  consider  the  influence  of  this  difference  upon  the  quo- 
tient. 

9.  A  line  is  measured  with  a  chain  (100  links  each  1  ft.  long).     After- 
wards, it  is  found  that  the  chain  is  one  foot  too  long.     If  the  measured 
length  was  10.36  chains,  what  is  its  true  length  if  the  error  is  assumed 
to  be  distributed  through  the  chain?  Ans.  10.4636  chains. 

10.  A  line  is  measured  with  a  100-ft.  tape  and  found  to  be  723.36 
feet  long.     The  tape  is  afterwards  found  to  be  0.02  of  a  foot  short. 
What  is  the  true  length  of  the  line?  Ans.  723.22  ft. 

11.  A  certain  steel  tape  is  of  standard  length  at  62°  F.     A  tape  will 
expand  or  contract  sixty-five  ten  millionths  of  its  length  for  each 
Fahrenheit  degree  change  of  temperature.     A  line  is  measured  when 
the  temperature  of  the  tape  is  approximately  80°  and  found  to  be 
323.56  feet  long.     What  is  its  true  length?     Is  it  necessary  to  know 
the  nominal  or  standard  length  of  the  tape  to  solve  this  problem? 

Ans.  323.52  ft. 

12.  What  change  in  temperature  is  necessary  to  change  a  100-foot 
tape  by  0.01  of  a  foot,  or  1  in  10,000?  Ans.  15°.38 

13.  A  certain  100-foot  steel  tape,  standard  length  at  62°  F.,  is  used 
to  measure  from  the  monuments  (Fig.  102)  to  the  point  A,  in  a  line 


X  Monument 


so'-; 


7th 

St. 

1 

2 

3 

4 

5 

6 

8th 
St. 

Monument  X 


FIG.  102 

between  lots  2  and  3  extended,  when  the  temperature  is  40°  F.  As- 
suming that  the  map  distances  are  correct,  what  lengths  must  be 
measured  from  7th  street  and  8th  street  monuments  respectively  to 
locate  the  point  A,  the  monuments  being  in  the  center  lines  of  the 
streets?  Ans.  160.02;  280.04 

Show  that  if  x,  y,  and  z  are  small  that 

14.  (1  +  z)(l  +  y}  is  nearly  equal  to  1  +  x  +  y. 

15.  (1  +  x)l(\  +  y)  is  nearly  equal  to  1  +  x  —  y. 


VIII,  §  143]  SMALL   ERRORS  181 

16.  (1  +  x)(l  +  ?/)(l  +  z)  is  nearly  equal  to  1  +  x  +  y  +  z. 

17.  Show  that  (1  +  0.03)  (1  -  0.05)  =  0.98  nearly. 

18.  Compute  (1.04)(1.06)(0.95).  Ans.  1.05 

19.  Compute  (a)  1.03/1.02;  (6)  (1.03)(1.02). 

Ans.  (a)  1.01;  (6)  1.05 

20.  Compute  (a)  (1.03)  (0.98);  (6)  1.03/0.98. 

Ans.  (a)  1.01;  (b)  1.05 

21.  Draw  a  figure  (rectangle)  to  represent  (4.03)  (9.02)  and  indi- 
cate 4  X  0.02;  9  X  0.03;  0.03  X  0.02;  4X9. 

22.  Show  that  the  error  in  abc  due  to  errors  Aa,  Ab,  Ac  in  a,  b,  and  c 
respectively,  is  be  •  Aa  +  ac  •  Ab  +  ab  •  Ac. 

23.  Compute  2.01  X  4.02  X  3.02      Draw  a  figure  (parallelepiped) 
to   represent   this   product   and   indicate   3  X  4  X  .01;   2  X  3  X  .02; 
2  X  4  X  .02;  .01  X  .02  X  .02;  2X4X3.  Ans.  24.4 

143.  Data  derived  from  Measurements.  The  preceding 
results  apply  immediately  to  the  case  in  which  numbers  ob- 
tained by  measurement  are  stated  without  any  accompanying 
indication  of  the  probable  error. 

In  such  cases  it  is  understood  that  the  given  figures  are  all 
reliable,  i.  e.,  that  we  stop  writing  decimal  places  as  soon  as 
they  are  doubtful.  The  last  figure  written  down  should  be  as 
accurate  as  is  possible.  Then  the  error  will  surely  not  be 
more  than  5  in  the  next  place  past  the  last  one  actually  written. 

Thus,  if  a  certain  length  is  reported  to  be  2.54  ft.,  we  would 
understand  that  the  true  length  is  not  more  than  2.545  ft.,  and 
not  less  than  2.535  ft.  For  if  the  true  length  is  more  than  2.545 
ft.,  it  should  be  given  as  2.55  ft.;  and  so  on. 

It  may  happen  that  the  last  figure  written  down  is  0.  This 
means  that  that  place  is  reliable.  Thus,  to  say  that  a  given 
length  is  2.4  ft.  means  that  the  true  length  is  between  2.35  ft. 
and  2.45  ft.  But  to  say  that  a  given  length  is  2.40  ft.  means 
that  the  true  length  is  between  2.395  ft.  and  2.405  ft. 

In  computations  based  upon  numbers  obtained  by  measure- 
ment, these  facts  must  be  kept  in  mind,  and  the  result  of  any 


182  MATHEMATICS  [VIII,  §143 

calculation  should  not  be  stated  to  more  decimal  places  than 
are  known  to  be  reliable. 

EXAMPLE  1.  Find  the  area  of  a  rectangle  whose  sides  are  found, 
by  actual  measurement,  to  be  2.54  ft.  and  6.24  ft.,  respectively. 

Since  the  error  in  writing  2.54  ft.  may  be  as  great  as  .005,  we  must 
write  for  the  length  of  this  side  (2.54  ±  .005)  ft.  Likewise,  we  must 
write  for  the  other  side  (6.24  ±  .005)  ft.  Hence,  by  the  rule  of  §  141, 
the  error  in  the  product  may  be  as  large  as 

2.54  X  .005  +  6.24  X  .005, 

that  is  .043.  Hence  we  are  not  justified  in  expressing  the  answer  to 
more  than  one  decimal  place;  although 

2.54  X  6.24  =  15.8496, 

we  must  sacrifice  all  the  figures  past  15.8,  and  write 
2.54  X  6.24  =  15.8  ±  .1 

since  the  true  answer  may  be  as  large  as  15.894  Even  the  figure  8 
in  the  first  decimal  place  is  not  reliable,  since  the  true  area  may  be 
nearer  15.9  than  15.8  sq.  ft. 

EXERCISES 

1.  Assuming   that   the  numbers   stated  below  are   the  results  of 
measurements,  and  that  each  of  them  is  stated  to  the  nearest  figure 
in  the  last  place,  find  the  required  answer  and  state  it  so  that  it  also  is 
correct  to  the  nearest  figure  in  the  last  place  you  give,  or  else  to  within 
a  stated  limit  of  possible  error. 

(a)  2.74  -f  3.48  +  11.25  +  7.34  Ans.  24.8 

(6)  3.25  -  7.348  +  4.26  -  6.1  Ans.  20.9  ±  .1 

(c)  6.27  X  3.14  (g)  61.54  X  45.2  +  14.81 

(d)  26.5  X  11.4  (A)  8.26  -=-  2.14 

(e)  7.32  X  5.4  (i)   43.7  +  5.4 

(/)  36.4  X  2.78  0')    (6-42  X  2.35)  -?-  4.5 

2.  The  sides  of  a  rectangle  are  measured,  and  are  found  to  be  4  ft. 
6.3  in.  by  3  ft.  5.4  in.     Express  correctly  the  area  of  the  rectangle. 

3.  The  three  sides  of  a  rectangular  block  are  measured  and  are 
found  to  be  7.4  in.  by  3.6  in.  by  4.7  in.     Express  the  volume. 


VIII,  §  146]  SMALL  ERRORS  183 

4.  Suppose  that  the  dimensions  of  a  bin  are  measured  roughly  to 
the  nearest  foot,  and  that  they  are  8  ft.  by  4  ft.  by  3  ft.     How  large 
may  the  volume  actually  be?     How  small  may  it  be? 

Ans.    118.1  cu.  ft.,  65.6  cu.  ft. 

5.  The  floor  of  a  room  is  found  by  measurement  to  be  22  ft.  X  15  ft., 
each  dimension  being  to  the  nearest  foot.     How  should  the  area  be 
stated?  Ans.    330  ±  18  sq.  ft.,  or  300  sq.  ft. 

6.  If,  in  Ex.  5,  the  height  of  the  room  is  9  ft.  to  within  the  nearest 
foot,  express  the  volume  of  the  room. 

144.  Error  in  a  Square.     If  a  is  an  observed  value  of  an  un- 
known quantity  u,  then  it  follows  directly  from  §  141  that  a 
measure  of  the  error  in  w2  is  approximately 

aAw  +  aAw  —  2aAw,  and  we  write 

w2  =  a2  ±  2aAw. 

That  is,  a  measure  of  the  error  in  the  square  of  a  reading  is  twice 
the  reading  times  its  error. 

145.  Error  in  a  Square  Root.     With  the  same  notation  as 

above,  u  =  a  ±  Aw  is  nearly  equal  to 

— 2 

"la"' 

since  the  last  term  is  small.     This  is  a  perfect  square  and  hence 
the  positive  square  root  of  w  is  approximately 

Va±-^=. 
2V  a' 

That  is,  a  measure  of  the  error  in  the  positive  square  root  of  a  read- 
ing is  equal  to  its  error  divided  by  twice  its  square  root. 

EXAMPLE.     Find  Vl25  ±  0.5      A  measure  of  the  error  is 

0.5/2(11.18)  =  .022     and     Vl25  ±  0.5  =  11.18  ±  0.022 
Again  V2400  =  V2401  -  1  =  49  -  &  =  49  -  0.0102 

146.  Errors  in  Trigonometric  Functions.     Suppose  a  ex- 
pressed in  radians  is  an  observed  value  of  an  unknown  angle  a. 


184  MATHEMATICS  [VIII,  146 

Then      a  =  a  ±  Aa  and  by  §  94, 

sin  a  =  sin  (a  ±  Aa)  =  sin  a  cos  Aa  ±  cos  a  sin  Aa. 
Now  if  Aa  is  small,  cos  Aa  is  nearly  equal  to  1,  and  sin  Aa  is  nearly 
equal  to  Aa.     Whence  we  have,  approximately, 
sin  a  •=  sin  a  ±  cos  a  •  Aa, 

and  the  smaller  Aa  is,  the  better  the  approximation.  Hence, 
a  measure  of  the  error  in  the  sine  of  an  angle  is  the  error  in  the 
reading  (expressed  in  radians)  multiplied  by  the  cosine  of  the 
reading. 

Similarly  we  can  show  that  a  measure  of  the  error  in  the  cosine 
of  an  angle  is  the  error  in  the  reading  multiplied  by  the  sine  of  the 
reading. 

By  means  of  these  results  and  the  principles  of  §  142  we  can 
readily  find  a  measure  of  the  error  in  the  other  trigonometric 
functions.  For  example 

_  sin  a  _  sin  (a  ±  Aa) 

tan  a  — — - ~ 

cos  a      cos  (a  ±  Aa) 

and  by  §  142,  a  measure  of  the  error  in  tan  a  is 

Aa(sin2  a  +  cos2  a)/cos2  a  =  sec2  a  •  Aa 

EXAMPLE,     sin  (36°  40'  ±  5')  =  sin  36°  40'  ±  .00145  cos  36°  40' 

=  .5972  ±  .0012 

cos  (36°  40'  ±  5')  =  cos  36°  40'  ±  .00145  sin  36°  40'  =  .8021  ±  .0009 
tan  (36°  40'  ±  5')  =  tan  36°  40'  ±  .00145  sec*  36°  40'  =  .7445  ±  .0023 

147.  Computation  of  Error  from  Tables.  This  will  be  illus- 
trated by  an  example.  To  find  sin  (36°  40'  ±  10')  we  look  in 
a  table  of  sines  and  find  sin  36°  50'  =  .5995,  sin  36°  40'  =  .5972, 
sin  36°  30'  =  .5948 ;  the  difference  between  the  first  and  second 
is  .0023  and  that  between  the  second  and  third  is  .0024.  Choos- 
ing the  larger  we  write  sin  (36°  40'  ±  10')  =  .5972  ±  .0024. 

This  method  applies  to  tables  of  logarithms,  squares,  square 
roots,  etc.,  in  fact  to  any  tables  giving  the  values  of  a  function 


VIII,  §  148] 


MATHEMATICS 


185 


In  practical 


4 


FIG.  103 


for  regularly  spaced  values  of  the  argument.  For  example,  a 
measure  of  the  error  in  log  u  =  log  (o  ±  Aw)  is  the  greater  of 
the  differences  log  (a  +  Aw)  —  log  a  and  log  a  —  log  (a  —  Aw). 
Thus  to  find  log  (17.4  ±  0.7)  we  look  up  in  the  table  log  16.7  = 
1.2227,  log  17.4  =  1.2405,  log  18.1  =  1.2577.  The  larger 
difference  is  0.0178  and  we  write 

log  (17.4  ±  0.7)  =  1.2405  ±  0.0178 
148.  Errors  in  Computed  Parts  of  Triangles, 
applications,    e.g.    in    surveying, 
the  given  parts  of  triangles  are 
subject    to    errors    of    measure- 
ment and  consequently  the  com- 
puted  parts   are   also    in   error. 
Suppose  the  base  AB  of  the  tri- 
angle ABC  in  Fig.  103  is  23.4  ± 
0.02,  the  side  AC  =  15.6  ±  0.04,  and  the  angle  A  =  32°  30'  ± 
10'.     Then  the  altitude 

CD  =  (15.6  ±  0.04)  sin  (32°  30'  ±  10') 

=  (15.6  ±  0.04) (0.5373  ±  0.0025)  §§  146,  147 

=  8.382  ±  0.060  §  141 

Again,  the  area  is  given  by  the  following  computation. 

Area  =  £(23.4  ±  0.02) (8.382  ±  0.06)  =  98.069  ±  0.786. 
Similarly  a  measure  of  the  error  in  any  computed  part  of  a 
triangle  may  be  found  by  the  foregoing  principles  of  this  chapter. 

EXERCISES 

Calculate  the  error  and  the  per  cent,  error  of  the  square  in  each  of 
the  following  numbers.  Where  no  estimate  of  the  error  is  expressed 
the  error  is  supposed  to  be  not  greater  than  5  in  the  next  place  past 
the  last  one  written  (§  143). 

1.  a  =  76  ±  0.1  4.    a  =  432  ±  0.03 

2.  a  =  101  ±  0.4  5.   a  =  2.46 

3.  a  =  32  ±  0.04  6.   a  =  13.4 


186  MATHEMATICS  [VIII,  §148 

Find  the  error  and  the  per  cent,  error  in  the  square  root  of  each  of  the 
following: 

7.  121  ±  0.4  11.  216  ±  0.03 

8.  169  ±  0.5  12.  165  ±  0.2 

9.  144  ±  0.02  13.  43.7 
10.  625  ±  0.01  14.  6.45 

15.  Show  that  the  error  of  the  cube  of  a  ±  Aa  is  ±  3a2-Aa.     Hence 
find  a  correct  expression  for  the  volume  of  a  cube  of  side  2.6  ft. 

16.  Show  that  the  error  of  the  fourth  power  of  a  ±  Aa  is  ±  4o3-Aa. 

17.  Show  that  the  error  of  the  cube  root  of  a  ±  Aa  is  Aa/3a2/3. 

18.  Find  by  the  use  of  the  tables  and  by  use  of  the  results  of  Ex.  17 
the  error  in  the  cube  root  of  (a)  1728  ±2;  (6)  15625  ±1;  (c)  343  ±  0.2 

Ans.  .005;  .0006;  .014 

19.  By  applying  twice  the  formula  for  the  error  of  the  .square  root 
of  a  ±  Aa,  show  that  the  error  of  the  fourth  root  of  a  ±  Aa  is  Aa/4a. 
Find  the  error  in  the  fourth  root  of  256  ±  1.  Ans.  0.001 

20.  Find  the  error  by  both  methods  of  sin  a  for  each  of  the  following: 

(a)  26°  ±  10'.  (6)  45°  ±  15'. 

(c)   80°  ±  30'.  (d)  10°  db  10'. 

Ans.  .0026;  .0031;  .0015;  .0028 

21.  Find  the  error  of  (a)  cos  a;  (6)  tan  «;  (c)  ctn  a;  (d)  sec  a;  (e) 
esc  a  due  to  an  error  Aa  in  a. 

22.  Find  by  the  use  of  the  tables  the  error  of  (a)  cos  (26°  db  25') ; 
(6)  tan  (20°  ±  3');  (c)  ctn  (70°  ±  20');  (d)  sec  (24°  ±  10');  (e)  esc  (46° 
±  10'). 

Ans.  (a)  .0032;  (b)  .0009;  (c)  .0066;  (d)  .0014;  (e)  .0039 

23.  Find  the  error  of  the  area  of  the  triangle  for  each  of  the  following : 

(a)  a  =  120  db  0.3  rod,  6  =  144  ±  0.2  rod,  y  =  47°  ±  10'. 

(6)  a  =  80  ±  0.1  rod,  b  =  160  ±  0.5  rod,  y  =  89°  ±  30'. 

(c)  a  =  40  ±  0.5  rod,  6  =    60  ±  0.3  rod,  y  =  45°  d=  10'. 

(d)  a  =  32  ±  0.4  rod,  b  =  146  ±  0.8  rod,  y  =  26°  ±5'. 

24.  If  A,  B,  C  denote  the  angles  and  a,  b,  c  the  sides  opposite  in  a 
plane  triangle  and  if  a,  A,  B  are  known  by  measurement,  then 

b  =  a  sin  B/sin  A. 


VIII,  §148]  SMALL   ERRORS  187 

Show  that  the  error,  called  the  partial  error  in  b  due  to  a  (written  A06), 
in  the  computed  value  of  b  due  to  an  error  Aa  in  measuring  a  is,  approxi- 
mately, 

A0&  =  sin  B  •  esc  A  •  Aa. 
Likewise  show  that 

AAb  =  —  a-sin  B-csc  A-ctn  A-&A,  and  Agb  =  a  cos  B-csc  A-&B, 
and  that  the  total  error  is,  approximately, 

A6  =  Aa&  +  AAb  +  AB&. 

Note  that  A  and  B  are  to  be  expressed  in  radian  measure. 

25.  The  measured  parts  of  a  triangle  and  their  probable  errors  are 

a  =  100  ±  0.01  ft.;        A  =  100°  ±  1';        B  =  40°  ±  1'. 
Show  that  the  partial  errors  in  the  side  b  are 

AQb  =  ±  0.007  ft.;        AAb  =  ±  0.003  ft.;        ABb  =  ±  0.023  ft. 

If  these  should  all  combine  with  like  signs,  the  maximum  total  error 
would  be  A6  =  ±  0.033  ft. 

26.  If  a  =  100  ft.,  B  =  40°,  A  =  80°,  and  each  is  subject  to  an  error 
of  1  %,  find  the  per  cent,  of  error  in  b. 

27.  Find  the  partial  and  total  errors  in  angle  B,  when 

a  =  100  db  0.01  ft.,        b  =  159  ±  0.01  ft.,        A  =  30°  ±  10'. 

28.  The  radius  of  the  base  and  the  altitude  of  a  right  circular  cone 
being  measured  to  1%,  what  is  the  possible  per  cent,  of  error  in  the 
volume?  Ans.  3%. 

29.  The  formula  for  index  of  refraction  is  m  =  sin  i/sin  r,  where  i 
denotes  the  angle  of  incidence,  and  r  the  angle  of  refraction.     If  i  =  50° 
and  r  =  40°,  each  subject  to  an  error  of  1%,  what  is  m,  and  what  its 
actual  and  percentage  error? 

30.  Water  is  flowing  through  a  pipe  of  length  L  ft.,  and  diameter 
D  ft.,  under  a  head  of  //  ft.     The  flow  in  cubic  feet  per  minute,  is 


Q  =  2356  J-  * 


IL  +  30D 

If  L  =  1000,  D  =  2,  and  H  =  100,  determine  the  change  in  Q  due  to 
an  increase  of  1%  in  H;  in  L;  in  D. 

31.  The  formula  for  the  area  of  a  triangle  hi  terms  of  its  three  sides 


188  MATHEMATICS  [VIII,  §148 


is  A  =  Vs(s  -  a)(s  -  b)(s  —  c)  where  s  =  \(a  +  b  +  c).  A  tri- 
angular field  is  measured  with  a  chain  that  is  afterwards  found  to  be 
one  link  too  long.  The  sides  as  measured  are  6  chains,  4  chains,  and 
3  chains  respectively.  What  is  the  computed  area,  and  what  is  the 
true  area? 

32.  Show  that  the  erroneous  area  of  a  field,  determined  from  measure- 
ments with  an  erroneous  tape,  will  be  to  the  true  area  as  the  square  of 
the  nominal  length  of  the  tape  is  to  the  square  of  its  true  length. 

33.  An  irregular  field  is  measured  with  a  chain  three  links  short. 
The  area  is  found  to  be  36.472  acres.     What  is  the  true  area? 

34.  The   acceleration   of   gravity   as   determined  by   an   Atwood's 
machine  is  given  by  the  formula:   g  =  2s /I2.     Find  approximately  the 
error  due  to  small  errors  in  observing  s  and  t. 

Ans.  A-sg  =  2As/P;  Atg  =  -  4s/l3. 

35.  A  right  circular  cylinder  has  an  altitude  12  ft.  and  the  radius 
of  its  base  is  3  ft.     Find  the  change  in  its  volume  (a)  by  increasing 
the  altitude  by  0.1  ft.,  and  (6)  the  radius  by  0.01  ft.     (c)  By  increasing 
each  simultaneously.  Ans.  (a)  2.83;  (6)  3.02;  (c)  5.85 

36.  The  period  of  a  simple  pendulum  is 


.2,  jr. 

\<7 


9 

Show  that  AT  IT  =  %Al/l  —  %Ag/g  and  hence  a  small  positive  error 
of  k  per  cent,  in  observing  I  will  increase  the  computed  time  by  k/2%, 
and  a  small  positive  error  of  k'%  m  the  value  of  g  will  decrease  the 
computed  time  by  k'/2  per  cent. 

37.  Let  Wi  denote  the  weight  of  a  body  in  air,  and  w-i  its  weight  in 
water;  then  the  formula 

o   — 

Wi    —  Wz 

gives  the  specific  gravity  of  a  body  which  sinks  in  water.     If 
wi  =  16.5  ±  0.01,         w>2  =  12.3  ±  0.02, 

find  the  error  in  S  due  to  the  error  in  w\;  due  to  the  error  in  w2;  the 
total  error  in  S;  the  relative  error  AS/S. 

38.  The  specific  gravity  S  of  a  floating  body  is  given  by  the  expression 

S=    ^—  > 


VIII,  §148]  SMALL  ERRORS  189 

where  Wi  is  the  weight  of  the  body  in  air,  w2  is  the  weight  of  a  sinker 
in  water,  and  w3  is  the  weight  in  water  of  the  body  with  sinker  attached. 
Determine  the  specific  gravity  of  a  body  and  the  probable  error  if 

wi  =  16.5  ±  0.01 

wz  =  182.2  ±  0.03 

wa  =  176.5  ±  0.02  [RIETZ  AND  CRATHORNE] 

39.  To  determine  the  contents  of  a  silo  I  measure  the  inside  diameter 
and  height  in  feet  and  inches  and  find  D  =  8  ft.  2  in.,  h  =  21  ft.  6  in. 
Find  the  error  in  the  computed  contents  if  there  are  errors  AD  =  ±  0.4 
in.,  A/i  =  0.3  in.  in  the  measured  dimensions.  Ans.  2.22  cu.  ft. 

40.  My  neighbor  wants  to  buy  the  wheat  from  one  of  my  bins. 
The  measurements  are:  length  =  12  feet;  width  =  6  feet;  depth  of 
wheat  in  bin  =  8  ft.     I  make  a  mistake  however  of  1  /4  inch  in  measur- 
ing each  2  feet  of  linear  measure.     Find  the  error  of  contents  in  cubic 
inches.     Find  the  error  in  bushels  if  2150.4  cu.  in.  make  1  bushel. 
A  more  accurate  value  is  2150.42     Find  the  error  due  to  using  2150.4 
instead  of  2150.42     Find  the  error  if  2150  is  used. 

41.  I  decide  to  sell  to  a  neighbor  by  measurement  my  corn  in  the 
crib.     I  measure  with  a  yard  stick  placing  my  thumb  to  mark  the 
end  of  the  yard  and  holding  my  thumb  in  place  proceed  to  measure 
beyond  it  thus  making  an  error  of  1/2  inch.     My  measurements  are 
length  =  30  ft.  3  in.;  width  =  11  ft.  9  in.;  height  13  ft.  6  in.     Find  the 
error  in  cubic  inches  due  to  my  method  of  measuring. 

42.  The  quantity  of  water  in  cubic  feet  per  second  flowing  through 
a  rectangular  weir  is  given  by  the  formula. 

Q  =*  3.33  [L  -  2h]hw, 

where  h  is  the  depth  in  feet  of  water  over  the  sill  of  the  weir,  and  L 
the  length  in  feet  of  the  sill.  Find  Q  and  the  error  hi  Q  if  L  =  26  ±  0.1, 
h  =  1.6  ±  0.02 


CHAPTER  IX 

CONIC   SECTIONS 

149.  Derivation.     The  circle,  the  ellipse,  the  parabola,  and 
the  hyperbola,  are  curves  which  can  be  cut  out  of  a  right  circular 
conical  surface  by  planes  passing  through  it  in  various  directions. 
For  this  reason,   they  are   called  also   conic  sections.     Being 
plane  curves,  however,  they  can  be  defined  and  studied  as  the 
locus  of  a  point  moving  in  a  plane  under  certain  conditions. 

150.  The  Circle.     The  circle  is  the  locus  of  a  point  moving  at 
a  fixed  distance  r  from  a  fixed 

point  C. 

The  fixed  distance  r  is  called 
the  radius;  the  fixed  point  C  is 
called  the  center. 

EQUATION  OF  THE  CIRCLE. 
Given  the  center,  C(x0,  y0)  and 
the  radius,  r,  of  a  circle,  to  de- 
duce its  equation. 

Let  P(x,  y)  be  any  point  on  the  locus  (Fig.  104).     Then  by 

(D§  45,  

CP  =  V(x  —  z0)2  +  (y  —  2/o)2, 

and  by  the  definition  of  the  circle  CP  =  r.     Hence,  squaring 
and  equating  the  two  values  of  CP  ,  we  find 

(1)  (x  -  x0)2  +  (y  -  y0Y  =  r\ 

Conversely,  let  Q(x\,  y\)  be  any  point  which  satisfies  (1);  i.  e., 


~7 


FIG.  104 


-  x0)2  +  (t/i  -  y0)»  = 
190 


IX,  §152]  CONIC  SECTIONS  191 

whence 


(*i  -  so)2  +  (</!  -  yoy  =  r, 

but  this  says  that  CQ  =  r,  and  therefore  Q  is  on  the  circle. 
Therefore  (1)  is  the  equation  of  the  circle. 

If  the  center  is  at  the  origin,  x0  =  yo  =  0,  and  the  equation 
reduces  to 

(2)  x2  +  y2  =  r2. 

151.  Equation  of  the  Second  Degree.     The  most  general 
equation  of  the  second  degree  in  x  and  y  is  of  the  form 

(3)  a.-c2  +  bxy  +  cy2  +  dx  +  cy  +  f  =  0, 

in  which  the  coefficients  are  real  numbers  and  a,  6,  c,  are  not 
all  zero.  The  equation  of  the  circle  which  we  have  obtained 
is  of  this  form  and  has  always  6  =  0  and  a  =  c.  Conversely, 
the  special  equation  of  the  second  degree 

(4)  ox2  +  ay2  +  dx  +  ey  +  f  =  0. 

is  the  equation  of  a  circle  or  of  no  locus.  To  show  this  we 
have  only  to  complete  the  square  of  the  terms  in  x  and  of  the 
terms  in  y.  This  process  will  reduce  it  to  the  form  of  (1)  §  150, 
as  is  shown  in  the  next  paragraph. 

152.  Determination  of  Center  and  Radius.     When  the 
equation  of  a  circle  is  given,  the  center  and  radius  can  be  found 
by  transposing  the  constant  term  to  the  right  and  completing 
the  square  of  the  terms  in  x  and  also  of  the  terms  in  y. 

EXAMPLE  1.     Find  the  center  and  radius  of  the  circle 

x2  +  y*  -  3x  -  2y  -  3  =  0. 
To  reduce  this  equation  to  the  form  (1)  we  complete  the  squares  as 

follows: 

(z2  -  3x  +    )  +  (y2  -  2y  +    )  =  3, 

(z2  -  3x  +  |)  +  &  -  2y  +  1)  =  3  +  |  +  1, 

(x  -  f)2  +  (y  -  I)2  =  (f)2 


192  MATHEMATICS  [IX,  §  152 

Comparing  this  with  the  standard  equation  (1),  we  see  that  the  center 
is  at  (3/2,  1)  and  r  =  5/2. 

EXAMPLE  2.     Examine  the  equation 

9x2  +  9?/2  -  6x  +  12y  +  6  =  0, 
We  complete  the  squares  as  follows : 

&  +  y2  -  lx  +  f  y  +  f  =  0, 
X*  -  \x  +  $  +  2/2  +  f  y  +  f  =  -  f  +  i  +  f, 
(s  ~  i)2  +  (2/  +  §)2  =  ~  i 

But  since  the  square  of  a  real  number  is  positive  (or  zero),  this  shows 
that  there  are  no  points  in  the  plane  which  satisfy  the  given  equation. 
Therefore  it  has  no  locus. 

EXAMPLE  3.     Examine  the  equation 

225x2  +  225?/2  -  270x  -  300t/  +  181  =  0. 
We  complete  the  squares  as  follows: 

x2  +  y2  -  fx  -  fa  +  iH  =  0, 
x2  -  fx  +  A  +  2/2  -  f !/  +  I  =  -  Mi  +  &  +  *, 

(x  -  f  )2  +  (y  -  I)2  =  o. 

This  shows  that  the  given  equation  is  satisfied  by  the  point  (3/5,  2/3) 
and  by  no  other  point  in  the  plane.  This  case  may  be  looked  upon  as 
the  limiting  case  of  a  circle  whose  center  is  at  (3/5,  2/3),  and  whose 
radius  is  zero. 

EXERCISES 

1.   Write  the  equation  of  the  circle  determined  by  each  of  the  follow- 
ing conditions. 

(0)  Center  (2,  4),  radius  =  3.  (6)  Center  (—  1,  3),  radius  =  5. 

(c)  Center  (-2,  -3),  radius  =  3.  (d)  Center  (3,  -  2),  diameter  =  7. 
(e)  Center  (a,  a),  diameter  a.  (f)  Center  (r,  0),  radius  =  r. 

(g)  Center  (4,  6)  passes  through  the  point  (0,  3). 
(h)  Abscissa  of  center  =  1,  passes  through  the  points  (0,  —  1),  (0,  7). 

(1)  The  segment  from  (1,  —  3)  to  (7,  5)  is  a  diameter. 

0)    Center  is  on  the  line  x  =  y,  tangent  to  x-axis  at  (—  6,  0). 


IX,  §  152]  CONIC   SECTIONS  193 

2.  Write  the  equation  of  a  circle  of  radius  6  when  the  origin  is  (a)  at 
the  highest  point  of  the  circle ;  (6)  at  the  lowest  point ;    (c)  at  the  left- 
most point ;    (d)  at  the  rightmost  point ;    (e)  when  the  origin  divides 
the  horizontal  diameter  from  left  to  right  in  the  ratio  1/3. 

3.  Determine  which  of  the  following  equations  represent  circles; 
find  the  center  and  the  radius  in  each  case. 

(a)  x2  +  y2  =  4x.  (6)  x2  +  y2  =  6y. 

(c)  x2  +  8y  =  4x  -  y2.  (d)  3z2  +  3y2  =  14y. 

(e)  x*  +  y2  +  4x  +  7  =  0.  (/)    x2  +  y2  +  3x  +  5y  =  0. 

(0)  x2  +  y2  =  2(y  +  4).  (h)  x2  +  y2  =  4(x  -  2). 

(1)  x2  +  y2  -  4x  -  6y  +  9  =  0. 
0')  *2  +  y2  +  101  =  87y  -  20x. 
(A;)  2z2  +  27/2  +  15y  =  12x  +  7. 
(1)  9x2  +  9y2  +  6y  =  24x  +  47. 
(TO)  16.x2  +  167/2  =  24x  +  40y  -  34. 
(n)  49x2  +  49y2  +  28x  -  2%  +  9  =  0. 

(o)    4a(ax2  +  6x  -  by)  +  b2  +  4o(ay2  -  ex  -  cy)  +  c2  =  0. 

4.  Show  that  if  the  coefficients  of  x2  and  y2  in  the  equation  of  a 
circle  are  each  +  1,  the  coordinates  of  the  center  can  be  found  by 
taking  negative  one-half  the  coefficient  of  x  and  negative  one-half  the  coef- 
ficient of  y. 

For  example,  the  center  of  the  circle 

x2  +  y2  -  5x  +  4y  -  3  =  0 

is  (5/2,  -  2). 

5.  Find  the  coordinates  of  the  center  of  each  of  the  following  circles, 
by  the  process  of  Ex.  4. 

(a)  x2  +  y2  -  4x  -  6y  +  9  =  0.  (d)  x2  +  y2  -  2x  +  4y  +  1  =  0. 
(6)  x2  +  y2  +  6x  +  4y  +  9  =  0.  (e)  x2  +  y2  -  3x  +  5y  +  3  =  0. 
(c)  x2  +  ?/  -  4y  =  0.  (/)  2x2  +  2y2  +  4x  -  6y  +  1  =0. 

6.  The  value  of  the  polynomial  P  =  x2  +  y2  —  2x  —  4y  +  3  at  any 
point  of  the  xy-plane  is  found  by  substituting  the  coordinates  of  the 
point  for  r  and  y  in  P.     Thus  at  (3,  2),  P  =  2.     Show  that  all  points 
at  which  P  is  positive  lie  outside  a  certain  circle,  and  all  points  at 
which  P  is  negative  lie  inside  the  same  circle.     With  respect  to  this 
circle,  where  are  the  points  (0,  1),  (1,  2),  (2,  3),  (4,  5),  (0,  3),  (1,  4), 
(2,  2)? 

14 


194 


MATHEMATICS 


[IX,  §  153 


153.  Translation  of  Axes.  Given  a  pair  of  axes  OX  and 
0  Y,  a  curve  C,  and  its  equation  in  terms  of  the  coordinates 
x  =  OA  and  y  =  AP.  (Fig.  105.)  Move  the  origin  to  the 

point  0'  whose  coordinates 
referred  to  the  old  axes  are 
(h,  k}  and  draw  new  axes 
O'X'  and  O'Yf  parallel  to  the 
x  old  axes.  The  curve  is  not 
moved  or  changed  but  the 

Y 

-»-  coordinates  of  all  its  points 
are  changed,  and  its  equation 
is  changed. 


4-°- 


FIG.  105 


From  the  figure  we  see  that 


and 


x  =  x'  +  h 

y  =  y'  +  k. 


These  equations  are  true  no  matter  which  way  nor  how  far  the 
origin  is  moved  if  the  new  axes  are  parallel  to  the  old  ones. 
These  values  substituted  in  the  old  equation  of  the  curve, 
give  the  new  equation.  Hence,  to  find  the  new  equation, 
substitute  in  the  old  equation,  in  the  place  of  x,  the  new  x  plus 
the  abscissa  of  the  new  origin  and  in  the  place  of  y,  the  new  y  plus 
the  ordinate  of  the  new  origin. 

EXAMPLE.     Translate  the  origin  to  the  point  (1,  —  2)  on  the  circle 

3x2  +  3?/2  -  5x  +  2y  =  6. 
The  new  equation  is 

3(x'  +  I)2  +  3(y'  -  2)'  -  5(z'  +  1)  +  2(y'  -  2)  =  6, 
and  this  reduces  to 

3x'2  +  3y'2  +  x'  -  1<V  =  0. 


IX,  §155] 


CONIC  SECTIONS 


195 


154.  Parabola.     The  parabola  is  the  locus  of  a  point  which 
moves  so  as  to  be  always  equidistant  from  a  fixed  point  F  and  a 
fixed  line  L. 

The  fixed  point  F  is  called  the  focus.     The  fixed  line  L  is 
called  the  directrix. 

155.  Equation  of  the  Parabola.     Let  F  be  the  focus  and 
RS  the  directrix  of  a  parabola.     (Fig.   106.)     Draw  FD  per- 
pendicular to  the  directrix.     The 

midpoint  0  between  D  and  F 
is  on  the  parabola.  Take  0  for 
the  origin,  OF  for  the  re-axis, 
and  take  OY  parallel  to  the 
directrix  for  ?/-axis.  Let  the 
distance  DO  =  OF  =  p.  Then 
the  coordinates  of  the  focus 
are  (p,  o).  Let  P(x,  y)  be  any 
point  on  the  parabola.  By 
definition,  FP  =  NP;  but 


FIG.  106 


and 
whence 


FP  =     (z  -  p)2  +  y2, 
NP  =  x  +  p, 


Squaring  this,  we  find 
(5) 


p)2  +  2/2  =  x  +  p. 


=  ipx. 


We  have  now  proved  that  every  point  on  the  parabola  satis- 
fies the  equation  (5).  It  follows  that  the  parabola  has  no 
points  on  the  left  of  the  y-axis,  for  negative  values  of  x  cannot 
satisfy  the  equation  (5). 

Conversely,  let  PI(XI,  y\)  be  a  point  which  satisfies  (5);  then 


?/i2  =  4pzi,         and         (x\  —  p)2  =  (xi  —  p)2, 


196 


MATHEMATICS 


[IX,  §  155 


whence,  adding,  we  have 

(xi  -  p)2  +  yS  =  (xi  +  p)2, 
that  is 

FP?  =  N\P?. 

Therefore  PI  is  on  the  parabola.  This  completes  the  proof 
that  (5)  is  the  equation  of  the  parabola. 

The  parabola  is  symmetric  with  respect  to  the  line  through 
its  focus  perpendicular  to  its  directrix.  This  line  is  called  the 
axis  of  the  parabola.  The  point  where  the  parabola  crosses 
its  axis  is  called  its  vertex.  The  chord  through  the  focus 
perpendicular  to  the  axis  of  the  parabola  is  called  its  latus 
rectum.  Let  the  student  show  that  the  length  of  the  latus 
rectum  is  4p. 

The  parabola  y2  =  4px  crosses  every  horizontal  line  exactly 
once,  and  every  vertical  line  to  the  right  of  the  7/-axis  twice, 
once  above  and  once  below  the  z-axis.  The  farther  the  vertical 
line  is  to  the  right,  the  farther  from  the  z-axis  does  the  curve 
cut  it. 

By  analogy  to  (5)  it  is  evident  that  the  equations  of  the 
parabolas  shown  in  Figs.  107,  108,  109  are,  respectively, 


FIG.  107 


FIG.  109 


(6)  t/2  =  -  4pz,         (7)  x*  =  4py,         (8)  z2  =  -  4py. 

The  position  of  each  of  these  curves  should  be  related  to  its 
equation  as  follows:  yz  =  4px  is  a  parabola  tangent  to  the  y-axis 
at  the  origin,  having  its  focus  on  the  x-axis  to  the  right.  The 
student  should  make  similar  statements  concerning  equations 
(6),  (7),  and  (8). 


IX,  §156]  CONIC  SECTIONS  197 

156.  Vertex  not  at  the  Origin.     Each  of  the  equations 

(9)  (y  -  k)2  =  db  4p(z  -  h), 

(10)  (x  -  h)*  =  ±  4p(y  -  k) 

represents  a  parabola  whose  vertex  is  at  (h,  k)  and  whose  axis  is 
either  horizontal  (equation  (9))  or  vertical  (equation  (10)).     For, 
on  translating  the  axes  to  this  point  they  reduce  to  one  of 
the  types  (5),  (6),  (7),  or  (8)  considered  above. 
In  particular,  the  equation 

(11)  y  =  ax*+bx  +  c        (fl+0) 

represents  a  parabola  whose  axis  is  vertical.  It  is  concave  up 
or  down  according  as  a  is  positive  or  negative,  and  the  vertex, 
focus,  and  directrix  can  be  found  by  completing  the  square  of 
the  terms  in  x  and  reducing  it  to  the  form  (10). 

EXAMPLE  1.     Locate  the  parabola  y  =  2x2  —  8x  +  5.     Transposing, 

2x2  -  8x  =  y  -  5; 
dividing  by  2, 

x2  —  4x  =  \y  —  \  ; 
adding  4, 

x2  -  4x  +  4  =  \y  +  f  ; 


Hence  the  vertex  is  the  point  (2,  —  3),  and  p  =  |.  The  parabola  is 
concave  upwards;  its  focus  is  |  above  the  vertex,  and  its  directrix  is  £ 
below  the  vertex. 

EXAMPLE  2.     Examine  the  equation  y  =  —  2x2  +  4x.     We  may 
write  successively  the  equations 


x2-2x=-|y,         x2-2x  +  l  = 

Hence  the  vertex  is  at  the  point  (1,  2),  and  p  =  |.  The  parabola  is 
concave  downwards,  its  focus  is  £  below  the  vertex,  and  its  directrix 
is  j  above  the  vertex. 

Similarly,  the  equation  x  =  ay2  -\-by-\-c  can  be  reduced  to 
the  type  (9)  by  completing  the  square  of  the  terms  in  y,  and 
from  this  a  sketch  of  the  parabola  can  be  made. 


198  MATHEMATICS  [IX,  §  156 

EXERCISES 

1.  Sketch  each  of  the  following  parabolas,  write  the  equation  of  its 
directrix,  and  the  coordinates  of  its  focus  and  vertex: 

(a)  y*  =  Sx.  (d)   %2  =  3z.  (g)   (x  +  3)2  =  5(3  -  y). 

(b)  x*  =  Gy.  (e)    2y*  =  25z.  (h)  x2  =  I0(y  +  1). 

(c)  y2  =  -  3z.      (/)  (y  -  2)2  =  8(s  -  5).      (i)    (y  +  4)2  =  -  6z. 

2.  Sketch  each  of  the  following  parabolas,  and  find  the  coordinates 
of  the  vertex  and  focus  and  the  equations  of  the  directrix  and  axis. 

(a)  y2  -  2y  -  4x  +  6  =  0.  (b)   y*  +  ±y  -  6x  =  0. 

(c)   x2  +  4z  +  6y  -  8  =  0.  (d)   a;2  -  x  +  y  =  0. 

(e)  4z2  -  12x  +  3y  -  2  =  0.  (/)  3y2  +  6?/  -  7x  -  10  =  0. 

3.  Sketch  the  parabolas  with  the  following  lines  and  points  as  direc- 
trices and  foci,  respectively;  and  find  their  equations. 

(a)  x  -  3  =  0,     (6,  -  3).  (b)  x  =  0,     (-  2,  -  2). 

(c)   y  +  4  =  0,     (-  2,  0).  (d)  y  -  26  =  0,     (0,  0). 

4.  Find  the  parabolas  with  the  following  points  as  vertices  and  foci, 
respectively. 

(a)   (0,  0),  (2,  0).  (6)   (1,  1),  (3,  1). 

(c)    (-  2,  -  2),  (-  4,  -  2).  (d)  (3,  2),  (3,  6). 

5.  Find  the  parabola  with  vertex  at  the  origin  and  axis  parallel  to 
the  x-axis,  and  passing  through  the  point : 

(4,1);     (2,3);     (1,1);     (-1,2);     (2,  -  4);     (-  2,  -  5). 

6.  The  cable  of  a  suspension  bridge  assumes  the  shape  of  a  parabola 
if  the  weight  of  the  suspended  roadbed  (together  with  that  of  the  cables) 
is  uniformly  distributed  horizontally.     Suppose  the  towers  of  a  bridge 
240  ft.  long  are  60  ft.  high  and  the  lowest  point  of  the  cables  is  20  ft. 
above  the  roadway.     Find  the  vertical  distances  from  the  roadway  to 
the  cables  at  intervals  of  20  ft. 

7.  An  arch  in  the  form  of  a  parabolic  curve  is  29  ft.  across  the 
bottom  and  the  highest  point  is  8  ft.  above  the  horizontal.     What  is 
the  length  of  a  beam  placed  horizontally  across  it,  4  ft.  from  the  top? 

8.  A  parabolic  reflector  is  8  inches  across  and  8  inches  deep.     How 
far  is  the  focus  from  the  vertex?  Ans.  2  in. 


IX,  §157] 


CONIC  SECTIONS 


199 


157.  Ellipse.     An  ellipse  is  the  locus  of  a  point  which  moves 
so  that  the  sum  of  its  distances  from  two  fixed  points  is  constant. 

The  fixed  points  F  and  F'  (Fig.  110)  are  called  the  foci.  Let 
the  constant  distance  be  2a; 
this  cannot  be  less  than  F'F. 
If  it  is  just  equal  to  F'F  the 
locus  is  evidently  the  seg- 
ment F'F.  Hence  we  assume 
that  2a  >  F'F.  Take  the 

x-axis  through  the  foci,  and 

FIG.  110 
the   origin   midway   between 

them.     Then  for  all  positions  of  the  moving  point  P,  we  have 


0 


(12) 


F'P  +  FP  =  2a. 


One  position  of  P  is  a  certain  point  A  on  the  z-axis  to  the 
right  of  F,  and  by  (12), 


and 


F'A  +  FA  =  2a 
OA  =  $(F'A  +  FA)  =  a. 


Similarly  the  point  A'  to  the  left  of  F'  such  that  A'O  =  a,  is  a 
point  on  the  ellipse.  The  points  A  and  A'  are  called  the  vertices. 
The  segment  A' A  is  called  the  major  axis  of  the  ellipse. 

Another  position  of  P  is  a  point  B  on  the  ?/-axis  above  0  and 
OB  is  denoted  by  b.     By  (12),  we  have 

F'B  +  FB  =  2a, 

and  since  B  is  on  the  perpendicular  bisector  of  F'F, 
F'B  =  FB  =  a. 

Similarly,  the  point  B'  below  0  such  that  B'O  =  b,  is  a  point  on 
the  ellipse.  The  distance  B'B  is  called  the  minor  axis.  The 


200  MATHEMATICS  [IX,  §  157 

intersection  of  the  major  and  minor  axes  is  called  the  center  of 
the  ellipse. 

The  rectangle  formed  by  drawing  lines  perpendicular  to  the 
major  and  minor  axes  at  their  extremities  is  called  the  rectangle 
on  the  axes. 

Let  a  denote  the  acute  angle  OFB.  Then  cos  a  is  called  the 
eccentricity  of  the  ellipse,  and  is  denoted  by  e.  It  is  evident 
that  e  =  OF  JO  A.  Hence,  from  the  right  triangle  OFB,  we 
have 

62 

(13)         0  <  e  <  I          and  -  =  sin2  a  =  1  -  e2. 

a2 

Since  OF  =  ae  the  coordinates  of  the  foci  F  and  F'  are  (ae,  o) 
and  (—  ae,  o),  respectively. 

Then  for  all  positions  of  the  moving  point  P,  by  (12),  we  have 


(14)  V(z  +  ae}2  +  y2  +  V(z  -  ae)2  +  y2  =  la. 

Transposing  the  second  radical,  squaring,  and  reducing,  we  find 


(15)  V(z  -  ae)2  +  y2  =  FP  =  a  -  ex, 

which  is  the  right-hand  focal  radius. 

Similarly,  on  transposing  the  first  radical  in  (14),  we  obtain 
the  equation 


(16)  (a  +  ae)2  +  i/  =  F'P  =  a  +  ex, 

which  is  the  left-hand  focal  radius.     Squaring  either  (15)  or  (16) 
and  reducing,  we  find 

(17)  (1  -  e2)x2  +  y2  =  a2(l  -  e2), 
whence,  by  (13), 

-j.2  n/2 

a*  +  V  =  L 
We  have  now  proved  that  every  point  on  the  ellipse  satisfies 


IX,   §157] 


CONIC   SECTIONS 


201 


(18).  It  can  be  proved,  conversely,  that  every  point  which 
satisfies  (18)  is  on  the  ellipse.  Hence  we  may  state  the  fol- 
lowing theorem. 

The  equation  of  the  ellipse  whose  semi-major  axis  is  a,  whose 
semi-minor  axis  is  b,  whose  center  is  at  the  origin,  and  whose  foci 
are  on  the  x-axis,  is 


(19) 


W 


The  numbers  a,  b,  e,  are  positive,  a  >  6,  e  <  1,  &2/a2  =  1  —  e2. 
The  coordinates  of  the  foci  are  (ae,  o)  and  (—  ae,  0).  The  focal 
distances  of  any  point  on  the  ellipse  are  a  —  ex  and  a  +  ex, 
respectively. 

The  equation  shows  that  the  curve  is  symmetric  with  respect 
to  the  x-axis  and  also  with  respect  to  the  ?/-axis.  It  follows 
that  the  curve  is  symmetric  with  respect  to  the  origin.  It  is 
only  necessary  to  plot  that  part  of  the  curve  which  lies  in  the 
first  quadrant  to  determine  the  shape  of  the  whole  curve,  which 
is  as  shown  in  Fig.  111. 


FIG.  112 


The  ellipse  can  be  drawn  by  the  continuous  motion  of  a  pencil 
point  by  means  of  a  pair  of  tacks  set  at  the  foci  and  a  loop  of 
string  around  them  as  shown  in  Fig.  112.  This1  is  the  best 
method  of  tracing  an  ellipse  on  a  drawing  board.  It  can  be 
used  to  lay  out  an  ellipse  of  any  desired  size  on  the  ground. 
Let  the  student  show  that  the  length  of  the  loop  of  string  is 
2o(l  +  e). 


202 


MATHEMATICS 


[IX,  §158 


158.  Auxiliary  Circle.     A  comparison  of  the  equation  of  the 
ellipse  (19)  with  that  of  the  circle 


(20) 


shows  that  any  ordinate  of  the  ellipse  is  to  the  corresponding 
ordinate  of  the  circle  as  b  is  to  a.  The 
diameter  of  this  circle  (20)  is  the 
major  axis  of  the  ellipse.  For  this 
reason,  the  circle  (20)  is  called  the 
major  auxiliary  circle,  or  simply  the 
auxiliary  circle.  The  points  where 
any  ordinate  cuts  the  ellipse  and  the 
auxiliary  circle  are  called  correspond- 
ing points. 


FIG.  113 


159.  Area  of  an  Ellipse.  Since  the  horizontal  dimensions 
of  the  ellipse  and  its  auxiliary  circle  are  the  same,  and  since 
their  vertical  dimensions  are  in  the  ratio  b  :  a,  we  have 


(21) 


Area  of  ellipse  6 

Area  of  auxiliary  circle       a 


Hence,  since  the  area  of  the  circle  is  known  to  be  iraz,  the  area, 
of  an  ellipse  whose  semi-axes  are  a  and  b  is  irab. 

160.  Projection.  If  a  circle  of 
radius  a  be  drawn  on  a  plane  making 
an  angle  a  with  the  horizontal  plane, 
then  the  vertical  projection  of  this 
circle  on  the  horizontal  plane  is  an 
ellipse  whose  semi-major  axis  is  a  and 
whose  semi-minor  axis  is  a  cos  a, 
since  its  ordinates  are  to  the  corre- 
sponding ordinates  of  the  circle  as  a  cos  a  is  to  a. 


FIG.  114 


IX,  §160]  CONIC  SECTIONS  203 

EXAMPLE  1.  Reduce  the  equation  of  the  ellipse  3x2  +  4y2  =  48  to 
standard  form;  find  a,  b,  and  c,  the  coordinates  of  the  foci,  the  focal 
distances  to  the  point  (2,  3),  and  the  area  of  the  ellipse. 

Dividing  through  by  48,  we  find 


Then,  by  comparison  with  (19),  we  have  o2  =  16  and  b2  =  12,  whence 
a  =  4  and  b  =  2V3.  From  (13)  we  find  e  =  £;  hence  ae  =  2.  It 
follows  that  the  foci  are  (—2,  0)  and  (2,  0).  The  right-hand  focal 
distance  to  (2,  3)  is  a  —  ex  =  3  and  the  left-hand  focal  distance  is 
a  +  ex  =  5.  The  area  is  irab  =  87rA/3  =  43.53  + 

EXAMPLE  2.     Reduce  the  equation  15x2  +  28y2  =  12. 

Dividing  by  12,  we  have 


_ 

4    "3  ' 


or 


Hence,  by  comparison  with  (19),  we  have  o  =  §VH  and  b  =  f  V21. 

EXERCISES 

1.  Find  the  semi-axes,  the  eccentricity,  locate  the  foci,  and  find  the 
focal  distances  to  any  point  (x,  y)  on  the  curve;  construct  the  rectangle 
on  the  axes,  and  sketch  the  curve: 

(a)  4x2  +    9i/2  =  36.  (b)     x*  +  25y*  =  100. 

(c)   9x2  +  25y2  =  225.  (d)   9x2  +  16?/2  =  144. 

(e)     x2  +    2y2  =  4.  (/)  6x2  +    9?/2  =  20. 

2.  In  each  of  the  following  cases  find  the  values  of  a,  b,  e,  if  they 
are  not  given.     Locate  the  foci,  and  write  the  equation  of  the  ellipse. 
Construct  the  rectangle  on  the  axes  and  sketch  the  curve. 


(a)   o  =  10,  b  =  6.  (g)  b  =  2^,  e  =  1/2. 

(6)   a  =  10,  6=8.  (h)  a  =  5,  e  =  2/3. 

(c)  o  =  5,  b  =  3.  (t)   a  =  6,  e  =  0. 

(d)  a  =  13,  e  =  12/13.  (j)   b  =  8,  e  =  3/5. 

(e)  a  =  7,  e  =  5/7.  (Jfc)  6  =  12,  e  =  5/13. 
(/)  a  =  10,  e  =  3/5.  (06=2,  e  =  1/3. 


204  MATHEMATICS  [IX,  §160 

3.  Find  the  area  of  each  of  the  ellipses  in  Ex.  1. 

4.  Show  that  any  oblique  plane  section  of  a  circular  cylinder  is  an 
ellipse. 

5.  Find  the  semi-axes  and  the  area  of  the  section  formed  by  cutting 
off  a  log  14  inches  in  diameter  by  a  plane  making  an  angle  of  60°  with 
its  length. 

6.  Design  a  flashing  (sheet  metal  collar)  for  a  four  inch  soil  pipe 
projecting  vertically  through  a  roof  whose  pitch  is  1/3. 

7.  A  circular  window  in  the  south  wall  of  a  building  is  4  ft.  in  diam- 
eter.    Light  from  the  sun  passes  through  the  window  and  falls  on  the 
floor.     Find  the  area  of  the  bright  spot  at  noon,  when  the  angle  of 
elevation  of  the  sun  is  (a)  60°,  (6)  45°,  (c)  30°. 

8.  An  ellipse  whose  semi-axes  are  10  and  9  is  in  a  horizontal  position. 
Through  what  angle  must  it  be  rotated  about  its  minor  axis  hi  order 
that  its  projection  on  a  horizontal  plane  shall  be  a  circle. 

Ans.  25°  50'. 

161.  Hyperbola.  A  hyperbola  is  the  locus  of  a  point  which 
moves  so  that  the  difference  of  its  distances  from  two  fixed  points  is 
constant. 

The  fixed  points  are  called  the  foci.  Other  terms  are  defined 
in  a  manner  analogous  to  those  for  the  ellipse. 

By  an  analysis  similar  to  that  given  in  §  157  for  the  ellipse, 
it  can  be  shown  that  the  equation  of  the  hyperbola  whose  semi- 
transverse  axis  is  a,  whose  semi-conjugate  axis  is  6,  whose 
center  is  at  the  origin  and  whose  foci  are  on  the  cc-axis,  is 


a*      V 

The  curve  consists  of  two  branches  and  is  symmetric  with 
respect  to  both  axes  and  with  respect  to  the  origin,  as  shown  in 
Fig.  115.  The  quantities  a,  b,  and  e  (=  sec  a),  are  positive, 
a  =  b,  e  >  1,  62/a2  =  e2  —  1;  the  coordinates  of  the  foci  are 
(ae,  o)  and  ( —  ae,  o) ;  the  focal  distances  to  a  point  on  the 
right  branch  are  ex  —  a  and  ex  +  a,  and  to  a  point  on  the 
left  branch,  the  negatives  of  these.  The  diagonals  OC  and  OC", 


IX,  §162] 


CONIC  SECTIONS 


205 


of  the  rectangle  on  the  axes  are  called  the  asymptotes  of  the 
hyperbola,   and   the   curve  approaches   nearer   and   nearer   to 


FIG.  115 

them  as  the  moving  point  recedes  from  the   vertices, 
equations  of  the  asymptotes  are 


The 


(23) 


and 


y-    --*. 


162.  Rectangular  or  Equilateral  Hyperbola.     If  the  semi- 
axes  of  a  hyperbola  are  equal,  b  =  a,  its  equation  reduces  to 
the  form 
(24)  x2  -  y2  =  a2. 

The  rectangle  on  the  axes  is  a  square,  the  eccentricity  is  sec  45° 
=  V2,  and  the  asymptotes  are  the  two  perpendicular  lines 
y  =  x  and  y  =  —  x.  This  is  called  a  rectangular  or  equilateral 
hyperbola.  It  plays  a  role  among  hyperbolas  analogous  to  that 
played  by  the  circle  among  ellipses. 

The  product  of  the  distances  of  any  point  on  an  equilateral 
hyperbola  to  its  asymptotes  is  constant.  For  the  distance  to 
the  asymptote  y  =  x  is  (x  —  y)  cos  45°,  and  the  distance  to 
the  asymptote  y  =  .—  x  is  (x  +  y)  cos  45°;  hence  the  product 
of  these  distances  is  a2  cos2  45°  =  £a2. 


206 


MATHEMATICS 


[IX,  §162 


It  follows  from  this  property  that  if  the  asymptotes  of  an 
equilateral  hyperbola  be  taken  for  coordinate  axes  the  equation 
of  the  curve  will  be 
(25)  xy  =  a  positive  constant, 

when  the  branches  are  in  the  first  and 
third  quadrants,  as  shown  in  Fig.  116; 
and  the  equation  will  be 

(26)        xy  =  a  negative  constant, 

when  the  branches  of  the  curve  are  in 
the  second  and  fourth  quadrants. 


FIG.  116 


EXAMPLE.     Reduce  the  equation  of  the  hyperbola  lQxz  —  9?/2  =  144 
to  standard  form. 

Dividing  by  144,  we  find 

i2  _  yL  = 

9        16 

Hence,  by  comparison  with  (22),  we  have  a  =  3,  b  =  4.     From  b?/a2 
=  e2  —  1  we  find  e  =  5/3. 

It  follows  that  the  coordinates  of  the  foci  are  (5,  0)  and  (—  5,  0). 
The  focal  distances  to  a  point  on  the  right  branch  are 

ex  —  a  =  l(5x  —  9)     and     ex  +  a  =  |(5a;  +  9). 

For  example  to  the  point  (6,  4V3)  they  are  7  and  13.     The  equations 
of  the  asymptotes  are 


\ 


rf 


A' 


y  =  fx     and     y  =  —  fz. 

To  sketch  the  curve,  lay  off  OA  =  3,  OB 
=  4,  Fig.  117,  construct  the  rectangle  on  the 
axes,  locate  the  foci  by  circumscribing  a  circle 
about  this  rectangle.  Sketch  in  the  curve 
free  hand  in  four  parts  beginning  each  time 
at  a  vertex,  using  the  asymptotes  as  guides, 
the  curve  approaching  them  in  distance  and  direction. 


FIG.  117 


IX,  §162]  CONIC   SECTIONS  207 

EXERCISES 

1.  Find  the  semi-axes,  the  eccentricity,  the  coordinates  of  the  foci, 
the  focal  distances  to  the  point  indicated,  the  equations  of  the  asymp- 
totes; construct  the  rectangle  on  the  axes  and  the  asymptotes,  and 
sketch  each  of  the  following  hyperbolas. 

(a)   4x2  -  9?/2  =  36,  (Vl3,  4/3). 

(6)   4x2-7/2  -8,  (-3/2,  1). 

(c)  3x2  -i/2  =  9,  (3,    -3>/2). 

(d)  3x2  -  4y2  =  1,  (-  V?;   V5). 

(e)  144x2  -  25y2  =  3600,  (10,    -  12  V3). 
(/)  9x2  -  16y2  =  576,  (12,  31/5). 

(g)  25x2  -  y2  =  100,  (-  V29,  25). 

(h)  225x2  -  647/2  =  14400,  (17,  28  J). 

(i)  x2  -  y2  =  9,  (-  5,  4). 

0')  x2  -  ?/2  =  400,  (101,  99). 

2.  Plot  on  the  same  axes  the  curves  xy  =  c,  for  c  =  1,  4,  6,  —  1, 
-  4,  -  6. 

3.  Find  the  equation  of  the  locus  of  a  point  which  moves  so  that 
the  difference  of  its  distances  from  the  two  points  (1,1)  and  (—  1,  —  1) 
is  constant  and  equal  to  2. 

4.  Find  the  locus  as  in  Ex.  3,  when  the  foci  are  (a,  a)  and  ( —  a,  —  a) 
and  the  constant  is  2a. 

5.  Find  the  locus  of  a  point  where  two  sounds  emitted  simultaneously 
at  intervals  one  second  apart  at  two  points  2,000  ft.  apart  are  heard  at 
the  same  time,  the  speed  of  sound  in  air  being  1,090  ft.  per  second. 

6.  On  a  level  plain  the  crack  of  a  rifle  and  the  thud  of  the  bullet 
on  the  target  are  heard  at  the  same  instant.     The  hearer  must  be  on  a 
certain  curve;  find  its  equation.     (Take  the  origin  midway  between  the 
marksman  and  the  target.) 

7.  By  translation  of  the  axes  (§  153)  find  the  equation  of  the  ellipse 
.(a)  whose  foci  are  (—  4,  2)  and  (0,  2),  and  whose  eccentricity  is  5. 

Ans.  3x2  +  4y2  +  12x  -  IQy  =  20. 
(6)  whose  vertices  are  (—2,  2)  and  (4,  2),  and  which  passes  through 

the  point  (1,4).  Ans.  4x2  +  9y2  -  8x  -  36?/  +  4  =  0. 

(c)   whose  semi-axes  are  5  and  3,  whose  right-hand  focus  is  at  (4,  —  4), 
and  whose  left-hand  vertex  at  (—  5,  —  4). 

Ans.  9x2  +  25r/2  +  200y  +  175  =  0. 


208  MATHEMATICS  [IX,  §162 

[HINT.  Start  with  the  equation  of  the  same  curve  when  its  center  is 
at  the  origin.] 

8.  By  the  method  of  Ex.  7,  find  the  equation  of  the  hyperbola 
(a)  whose  vertices  are  (—2,2)  and  (4,  2),  and  whose  eccentricity  is  5/3. 

Ans.  16x2  -  9?/2  -  32x  +  36y  =  164. 

(6)  whose  semiminor  axis  is  15,  whose  left-hand  vertex  is  at  (—  15,  3) 
and  whose  right-hand  focus  is  at  (10,  3). 

Ans.  225x2  -  64y2  +  3150x  +  384y  =  3951. 

(c)   which  passes  through  the  origin  and  whose  asymptotes  are  the 
lines  x  =  2  and  y  =  1.  Ans.  xy  =  x  +  y. 

163.  Intersection  of  Loci.  If  a  point  lies  on  a  curve,  its 
coordinates  must  satisfy  the  equation  of  that  curve.  Con- 
versely, any  pair  of  values  of  x  and  y  which  satisfy  an  equation 
determines  a  point  on  the  locus  of  that  equation.  If  the  same 
pair  of  values  of  x  and  y  satisfies  two  equations,  it  locates  a 
point  which  is  common  to  the  two  curves,  i.  e.,  a  point  of  inter- 
section. Hence,  to  find  the  points  of  intersection  of  two  curves, 
solve  their  equations  simultaneously  to  find  all  their  common 
solutions. 

EXAMPLE  1.  Find  the  intersections  of  the  line  3x  —  y  =  5  and  the 
ellipse  4x2  +  9?/2  =  25. 

Solving  the  first  equation  for  y  =  3x  —  5,  substituting  this  in  the 
second  and  reducing,  we  have 

17x2  -  54x  +  40  =  0. 
We  can  factor  this  quadratic  by  inspection : 

(17x  -  20)  (x  -  2)  =  0, 
whence 

xi  =  20/17    and    x2  =  2. 

Substituting  these  values  in  the  equation  3x  —  y  —  5,  gives  7/1  =  (—25/17) 
j/s  =1.  Therefore  the  points  of  intersection  are  (20/17,  —  25/17),  and 
(2,  1). 

Let  the  student  plot  the  curves  on  the  same  axes  and  verify  these 
results. 


IX,  §163]  CONIC  SECTIONS  209 

EXAMPLE  2.     Where  does  the  parabola 

3y  =  x2  -  5x  +  12 
intersect  the  ellipse 

4x2  +  3y2  =  48? 

Substituting  the  value  of  y  from  the  first  equation  in  the  second 
and  reducing,  we  get 

x4  -  lOx3  +  61x2  -  120x  =  0. 
Factoring  this  equation,  we  have 

x(x  -3)(x2  -  7x  +  40)  =  0 

and  we  see  by  inspection  that  Xi  =  0  and  x2  =  3  are  roots.     The  quad- 
ratic x2  —  7x  +  40  has  imaginary  roots. 

Substituting  these  values  of  x  in  the  first  given  equation  we  find 
?/i=4  and  7/2  =  2.     Hence  the  points  of  intersection  are  (0, 4)  and  (3, 2)- 

EXERCISES 

Find  the  points  of  intersection  of  the  following  pairs  of  curves. 

1.  x2  +  Qxy  +  9j/2  =4,     4x  +  3y  =  12. 

Ans.  (14/3,  -  20/9),  (10/3,  -  4/9). 

2.  x2  -  if  =0,     3x  -  2y  =  4.  Ans.  (4,  4),  (4/5,  -  4/5). 

3.  y2+x=0,     2y  +  x  =  0.  Ans.  (0,  0),  (-  4,  2). 

4.  x2  —  5y  =  0,     x  —  y  =  1. 

Ans.  x  =  i(5  ±  V5),  y  =  |(3  ±  V5). 

5.  2x  +  3y  =  5,     4x2  +  9(/2  +  16x  -  18^  -11=0. 

Ans.  (1,1),  (-2,3). 

6.  x  -  y  +  1  =  0,     (x  +  2)2  -  4y  =  0.  Ans.  (0,  1). 

7.  y  -  2x  =  0,     x2  +  y*  -  x  +  3y  =  0. 

Am.  (0,0),  (-  1,  -  2). 

8.  x  -  2y  +  4  =  0,     5x2  -  4y2  +  20  =  0.  Ans.  (I,  2£). 

9.  y  =  2x  -  3,     4y2  =  (x  +  3)(2x  -  3)«. 

Ans.  (3/2,  0),  (1,  -  1). 

10.  4i/2  =  x2(x  +  1),     y2  =  x(x  +  I)2.  Ans.  (0,  0),  (-  1,  0). 

11.  2x2  -  3i/2  =  -  58,     3x2  +  t/2  =  111. 

Ans.  (5,  6),  (-  5,  6),  (5,  -  6),  (-5,  -  6) 

12.  x2  =  4ay,     y  =  8n3/(x2  +  4oJ).  Ans.  (±  2a,  a). 
ITu  x2  +  y2  =  2,     x2  +  7/2  -  6x  -  6y  +  10  =  0.  Ans.  (1,  1). 

15 


210 


MATHEMATICS 


[IX,  §154 


164.  Straight  Line  and  Conic.  The  equations  of  the  circle, 
parabola,  ellipse,  and  hyperbola,  are  all  of  the  second  degree  in 
x  and  y.  Conversely,  it  can  be  shown  that  every  such  equa- 
tion represents  a  conic  section,  if  it  represents  any  curve  at  all. 
Given  a  straight  line  and  a  circle  we  know  that  one  of  three 
things  will  happen,  1)  there  may  be  two  intersections,  2)  there 
may  be  no  intersection,  or  3)  there  may  be  only  one  point  in 
common  and  then  the  line  is  a  tangent.  The  same  three  cases 
occur  with  the  intersections  of  a  straight  line  with  any  conic 
section.* 

When  we  solve  simultaneously  the  equation  of  a  straight  line 
with  the  equation  of  a  conic,  we  may  begin  by  substituting  the 
value  of  y  from  the  first  equation  in  the  second.  The  result  is 
a  quadratic  equation  in  x.  This  quadratic  equation  (§§  32,  33), 
(27)  Ax2  +  Ex  +  C  =  0 

will  have  1)  two  real  roots  when  B2  —  4 AC  >  0,  or  2)  no  real 
roots  when  Bz  —  4 AC  <  0,  or  3)  one  real  root  when  B2  —  4AC 

=  0.  These  algebraic  cases  cor- 
respond exactly  to  the  geometric 
cases  enumerated  above. 

EXAMPLE.  Of  the  three  parallel 
lines  8z  -  9y  =  20,  8z  -  Qy  =  30, 
and  8x  —  9y  =  25,  the  first  cuts 
the  ellipse  4x2  +  9?/2  =  25  in  two 
points  (5/2,  0)  and  (7/10,  -  8/5), 
the  second  does  not  intersect  it  at 
all,  and  the  third  intersects  it  at 
(2,  —  1)  only,  i.  e.  it  is  tangent  at 
that  point. 
The  resulting  quadratics  are,  respectively, 

20x2  -  64x  +    35  =  0,         B2  -  4AC  =  1296, 

.      20x2  -  96x  +  135  =  0,        B2  -  4AC  =  -  1584, 

x2  -    4x  +      4  =  0,        B*  -  4AC  =  0. 

*  There  is  one  exception  to  this  rule:  any  line  parallel  to  the  axis  of  a  parabola  h&t> 
one  and  only  one  point  in  common  with  the  curve,  but  no  such  line  is  a  tangent  to 
the  parabola. 


IX,  §165]  CONIC  SECTIONS  211 


1.  Show  that  one  of  the  three  lines  4x  +  25  =  Wy,  4x  +  27  =  lOy, 
4x  +  21  =  lOy,  intersects  the  parabola  y2  =  4x  in  two  points,  another 
is  tangent,  and  the  third  does  not  intersect  it  at  all. 

2.  Determine  whether  the  following  given  lines  are  tangent,  secant, 
or  do  not  meet,  the  corresponding  given  conic. 

(a)   x  +  y  +  1  =  0,  x2  =  4y. 

(6)    x  -  2y  +  20  =  0,  x2  +  y*  =  16. 

(c)  2x  +  3y  =8,  y2  =  4x. 

(d)  x  +  2y  =  5,  x2  +  y2  =  x  +  2y. 

(e)  2x  =  3y,  4x2  -  3y2  +  8x  =  16. 
(/)x  +  7/  =  8,  4x2+?/2  =  16x. 

3.  Find  the  points  in  which  the  circle  x2  +  y2  =  45  is  cut  by  the  lines 
(a)  2x  -  y  =  15,  (6)  2x  -  y  =  0,  (c)  2x  -  y  =  -  15. 

Ans.   (a)  (6,  -  3),   (6)  (3,  6),  (-  3,  -  6),   (c)  (-  6,  3). 

4.  Find  the  points  in  which  the  circle  x2  +  yz  —  6x  —  Qy  +  10  =  0 
is  cut  by  the  lines   (a)  x  +  y  =  2,   (6)  x  +  y  =  6,   (c)  x  +  y  =  10, 

(d)  x  -  y  =  0. 

Ans.  (a)  (1,  1),  (6)  (1,  5),  (5,  1),  (c)  (5,  5),  (d)  (1,  1),  (5,  5). 

5.  Find  the  points  in  which  the  parabola  3y  =  2x2  —  8x  +  6  is  cut 
by  the  lines  (a)  4x  +  3y  =  4,  (6)  4x  +  3y  =  6,  (c)  4x  +  3y  =  12. 

Ans.   (a)  (1,  0),   (6)  (0,  2),  (2,  -  f),  (c)  (3,  0),  (-  1,  5|). 

6.  Find  the  points  in  which  the  ellipse  3x2  +  4t/2  =  48  is  cut  by  the 
lines  (a)  x  +  2y  =  0,  (6)  x  +  2y  =  4,  (c)  x  +  2y  =  8,  (d)  x  +  2y  =  -4, 

(e)  x  +  2y  =  -_8. 

Ans.  (a)  (2\3,  -  V3),  (-2^3,  V3),  (b)  (4,  0),  (-  2,  3),  (c)  (2,  3), 
(rf)  (-4,0),  (2,  -3),    (e)  (-2,  -3). 

165.  Tangent    and    Normal.      Focal    Properties.      The 

equation  of  the  tangent  to  a  conic  can  be  found  by  the  principles 
of  §  164  if  the  slope  of  the  tangent  is  known,  or  if  the  coordinates 
of  one  point  on  the  tangent  are  known.  This  given  point  may 
be  the  point  of  contact  or  some  other  point  through  which  the 
tangent  is  to  pass. 

The  perpendicular  to  the  tangent  at  the  point  of  contact  is 
called  the  normal  to  the  conic  at  that  point.     When  the  slope 


212  MATHEMATICS  [IX,  §165 

of  the  tangent  is  known  or  can  be  found,  the  equation  of  the 
normal  can  be  written  by  the  principles  of  §§59  and  61. 

EXAMPLE  1.  Find  the  equation  of  the  tangent  to  the  parabola 
yz  =  24x  which  is  perpendicular  to  the  line  x  +  3y  +  1  =0. 

By  (13)  §  61,  the  slope  of  the  required  tangent  is  3,  and  by  (11)  §  59, 
y  =  3x  +  b  is  parallel  to  it  no  matter  what  value  b  has.  Proceeding 
to  find  the  points  where  this  line  intersects  the  parabola  we  are  led  to 
the  quadratic  equation, 

9z2  +  6(6  -  4)z  +  62  =  0. 

By  §  32,  this  quadratic  will  have  only  one  root  and  the  line  will  be 
tangent  to  the  parabola,  if 

36(6  -  4)2  -  3662  =  0. 

This  gives  6=2;  whence,  the  equation  of  the  required  tangent  is 
y  =  3x  +  2. 

EXAMPLE  2.  Find  the  equation  of  the  tangent  and  of  the  normal 
to  the  ellipse  3z2  +  4y2  =  48  at  the  point  (2,  3). 

We' first  verify  that  the  given  point  is  in  fact  on  the  ellipse.  Then 
by  (10)  §  59,  y  —  3  =  m  (x  —  2)  is  the  equation  of  a  line  through  (2,  3) 
no  matter  what  value  m  has.  Solving  this  simultaneously  with  the 
equation  of  the  ellipse  we  get  the  quadratic  equation, 

(4m2  +  3)z2  +  8m(3  -  2m)x  +  4(4m2  -  12m  -  3)  =  0. 

This  equation  will  have  only  one  root  and  the  line  will  be  tangent  to 
the  ellipse  if  (§32), 

64m2  (3  -  2m)2  -  16  (4m2  +  3)  (4m2  -  12m  -  3)  =0, 

that  is  if, 

4m2  +  4m  +  1  =0, 

whence  m  =  —  \  and  the  equation  of  the  required  tangent  is 

y  -  3  =  -  \(x  -  2),         or        x  +  2y  =  8, 

and  the  equation  of  the  normal  (whose  slope  by  (13)  §  61  is  2)  is 
y  -  3  =  2(x  -  2),         or        2x  -  y  =  1. 

The  normal  at  any  point  .P  on  a  parabola  bisects  the  angle 
between  the  focal  radius  FP,  and  the  line  through  P  parallel 
to  the  axis  of  the  curve. 


IX,  §165] 


CONIC   SECTIONS 


213 


We  learn  in  Physics  that  light  is  reflected  by  a  mirror  in  such 
a  way  that  the  angle  of  incidence  is  equal  to  the  angle  of  reflection. 
Hence,  a  ray  of  light  emanating  from  a  source  at  the  focus  and 


FIG.  119 


Fia.  120 


striking  the  parabola  at  any  point,  will  be  reflected  parallel  to 
the  axis.  This  is  the  principle  of  parabolic  reflectors  which 
are  extensively  used  for  head  lights.  It  is  easily  seen  that  if 
the  light  be  moved  slightly  beyond  the  focus,  the  reflected  rays 
will  tend  to  illuminate  the  axis. 

The  normal  at  any  point  of  an  ellipse  bisects  the  angle  between 
the  focal  radii  to  that  point,  Fig.  120.  It  follows  that  rays 
of  light,  or  sound,  emanating  from  one  focus  F,  will  after  re- 
flection by  the  ellipse,  converge  at  the  other  focus  F".  Hence 
the  name  focus.  This  is  the  principle  of  whispering  galleries. 

EXERCISES. 

1.  Find  the  equations  of  the  tangents  and  normals  to  the  following 
curves  at  the  points  indicated: 

(a)  y2  =  8x,         (2,  4),  (6)  x2  -  y2  =  64,         (10,  6), 

(c)   x2  +  3J/2  =  21,         (3,  -  2),       (d)  28?/2  =  27x,         (2J,  1J). 

2.  Find  the  equations  of  the  two  tangents  which  can  be  drawn  to 
the  parabola  y*  +  8x  =  0  from  the  point  (2,  1)  and  verify  that  they 
are  perpendicular. 

3.  Find  the  equations  of  the  tangent  and  normal  to  the  circle  x2  +  y* 
-  6x  -  Gy  +  10  =  0  at  the  point  (1,  1). 

Ans.  x  +  y  =  2,x  —  y  =  0. 


214  MATHEMATICS  [IX,  §  165 

4.  Find  the  equations  of  the  tangents  from  the  point  (9,  3)  to  the 
circle  x2  +  y2  —  45.  Ans.  2x  —  y  =  15,  x  +  2y  =  15. 

5.  Find  the  equations  of  the  tangent  and  normal  to  the  circle  xz  +  y2 
=  6z  +  2y  at  the  point  (2,  4). 

Ans.  x  -  3?/  +  10  =  0,  3x  +  y  =  10. 

6.  Find  the  equations  of  the  tangent  and  normal  to  the  hyperbola 
xy  =  6  at  the' point  (2,  3).      Ans.  3x  +  2y  =  12,  2x  -  3y  +  5  =  0. 

7.  Find  the  tangent  to  the  parabola  y2  =  12x  which  makes  an 
angle  of  60°  with  the  z-axis. 

8.  Find  the  tangent  to  the  parabola  y2  =  6x  which  makes  an  angle 
of  45°  with  the  re-axis. 

9.  Find  the  equations  of  the  tangents  to  the  circle 

(a)  x2  +  y2  =  4    parallel  to  2x  +  3y  +  1  =0, 

(b)  x2  +  y2  =  16  parallel  to  3x  -  2y  +  2  =  0. 

10.  Find  the  tangents  to  the  ellipse  Qx2  +  16?/2  =  144  which  make 
an  angle  of  30°  with  the  x-axis. 

11.  Find  the  equations  of  the  tangents  to  the  following  conies  which 
satisfy  the  condition  indicated. 

(a)  y2  =  4x,  slope  =  1/2.  (/)    x2  +  y2  =  25,  at  (4,  -  3). 

(6)  x2  +  y2  =  16,  slope  =  -  4/3.  (g)  x2  +  4?/2  =  8,    at  (-  2,  1). 

(c)  9z2  +  l&y2  =  144,  slope  =  -  1/4.  (h)     x2  -  y2  =  16,  at  (-  5,  3). 

(d)  x2  =  4y,  passing  through  (0,  -  1).  (i)   2y2  -  x2  =  4,    at  (2,  -  2). 

(e)  x2  =  Sy,  passing  through  (0,  2).  0')               2/2  =  %x>  at  (2»  16)- 

12.  Determine  the  condition  for  tangency  of  the  following  pairs  of 
curves. 

(a)  x2  —  y2  =  a2,     y  —  kx.  Ans.  k  =  ±  1. 

(6)  x2  +  y2  =  r2,     4y  -  3x  =  4Je.  Ans.  16/b2  =  25r2. 

(c)  4x2  +  y2  -  4x  -  8  =  0,     y  =  2x  +  k.  Ans.  k2  +  2k  -  17  =  0. 

(d)  xy  +  x  -  6  =  0,     x  =  ky  +  5.  Ans.  k2  +  14fc  +  25  =  0. 

13.  A  parabolic  reflector  is  12  inches  across  and  8  inches  deep. 
Where  is  the  focus? 

14.  The  ground  plan  of  an  auditorium  is  elliptic  in  shape.     The 
extreme  length  is  2,725  ft.  and  the  width  is  2,180  ft.     By  what  path 
will  a  sound  made  at  one  focus  arrive  first  at  the  other  focus,  i.  e.,  directly 
or  by  reflection  from  the  walls?     How  much  sooner  if  sound  travels 
1,090  ft.  per  second? 


IX,  §166] 


CONIC  SECTIONS 


215 


166.  Intersection  of  Conies.     Simultaneous  Quadratics. 

Two  conies  intersect,  in  general,  in  four  points.  Since  their 
equations  are  of  the  second  degree  in  x  and  y,  this  corresponds 
to  the  fact  that  two  quadratics  in  x  and  y  have,  in  general,  four 
solutions.  In  some  cases  these  solutions  are  not  all  real,  or 
there  may  be  less  than  four  so  that 
the  conies  represented  intersect  in 
less  than  four  points. 

As  shown  in  Fig.  121a,  the  hyper- 
bola x2  —  y-  =  5  intersects  the  ellipse 
x2  +  4?/2  =  25  in  the  four  points  (3, 2), 
(-  3,  2),  (-3,  -  2),  and  (3,  -  2). 
The  parabola  4x2  =  Qy  cuts  the  same 
ellipse  only  in  (3,  2)  and  (—  3,  2),  as 
shown  in  Fig.  1216. 

FIG.  121 

Certain  types  of  these  equations 

can  be  solved  by  elementary  methods.  The  most  important 
cases  will  now  be  explained. 

CASE  I.  When  all  the  terms  (except  the  constant  terms)  are 
of  the  second  degree  in  x  and  y. 

Eliminate  the  constant  terms  and  factor  the  result  into  two 
linear  factors. 

X2   -  7/2    =   5, 

x2  +  4?/2  =  25. 
Multiplying  the  first  equation  by  5  and  subtracting,  we  -have 

4x2  -  9y2  =  0, 
whence 

(2x  -  3y)(2x  +  3y)  =  0. 

Now  solving  simultaneously  the  two  pairs  of  equations 

\  2x  —  3y  =  0.  \  2x  +  By  =  0. 

We  find  that  the  solutions  of  (a)  are  (3,  2)  and  ( —  3,  —  2) ;  and  those 
of  (6)  are  (3,  —  2)  and  (—  3,  2).  It  is  easy  to  verify  that  these  are  all 
solutions  of  the  given  equations  by  actual  substitution. 


EXAMPLE  1 


•{ 


216  MATHEMATICS  [IX,  §166 

f  x2  +  3xy  =  28, 
EXAMPLE  2.    •<        ' 

t  4?/2  +  xy  =  8. 

Eliminating  the  absolute  terms,  we  have 

2X2  -  Xy  -  282/2  =  0, 
whence 

(2x  +  7y)(x  -  4y)  =  0. 


This  gives  the  two  pairs  of  simultaneous  equations 

(«\  S  42/2  +  W  =  8'         ™  /  4^2  +  sy  =  8, 
W    I  2x  +  7y  =  0,         W  1     x  -  4y  =  0. 

The  solutions  are  therefore  (14,  -  4),  (-  14,  4),  (4,  1),  (-  4,  -  1). 
Verify  each  of  these  by  actual  substitution. 

SPECIAL  METHOD,  CASE  I.  If  there  is  no  term  in  xy  the 
equations  can  be  solved  as  linear  equations  considering  z2 
and  i/2  as  the  unknowns. 


r  x2 

EXAMPLE.    •! 

t  x2  • 


-  y1  =  5, 

+  42/2  =  25. 

Eliminate  x2  and  solve  for  y2.  This  gives  y2  =  4,  whence  y  =  ±  2. 
Then  eliminate  y2  and  solve  for  x2.  This  gives  x2  =  9,  whence  x  =  ±  3. 
Verify  that  (3,  2),  (-  3,  2),  (-3,  -  2),  (3,  -  2),  are  all  solutions  of 
the  given  equations. 

CASE  II.  When  the  equations  are  symmetric  in  x  and  y; 
i.  e.,  when  the  interchange  of  x  and  y  leaves  the  equations 
unchanged. 

/•      2    _i_  q.2    "1  O 

EXAMPLE.    -\ 

L          xy  =  6. 

Substituting  s  +  t  =  x  and  s  —  t  =  y  in  the  given  equations,  we  find 
2s2  +  2£>  =  13, 

S2   _  ft    -  Qf 

Solving  these  equations,  we  have 

s  =  ±5/2,         t  =  ±1/2. 

Hence  the  values  of  x  and  y  are  x  =  ±  3  or  ±  2,  y  =  ±  3,  or  ±  2. 
Testing  these  values  in  the  given  equations  we  verify  that  (3,  2),  (2,  3) 
(—2,  —  3),  (—3,  —  2),  are  solutions. 


IX,  §  166]  CONIC  SECTIONS  217 


EXERCISES 

Solve  the  following  pairs  of  simultaneous  equations. 

J  4x2  +  4xy  -  y2  =  7x  -  y,  |  x2  -  y2  +  16  =  0, 

i  4x  +  3y  =  1.  1  (x  +  I)2  =  (y  +  I)2. 

f  3x2  +  4y2  =  48,  J  5x2  +  7y2  =  225, 

1  y2  =  3(1  -  x).  \  2x  +  3y  =  9. 

f  4x2  +  3xy  =  10,  f  a?  +  y2  =  153> 

I  3y2  +  4xy  =  20.  1  xy  =  36. 

J  2x2  +  2xt/  +  5y2  =  40,  f  x2  +  y2  -  x  -  y  =  204, 

1  x2  - 

f  2x2  +  xy  +  37/2  =  12,  x2  -  2y*  +  1  =  0, 


y2  +  2x  -  2y  =  0.  xy  +  x  +  y  =  129. 


, 
1  2x  +  y  =  0. 


y  =  0.  2x2  -  3y2  -  23  =  0. 

3x2  +  4?/2  =  48,  j  4x2  +  6xi/  +  4y2  =  46, 

\  X2  +  ^  =  34. 

x2  +  2xy  =  407,  f  x2  +  2?/2  =  123, 

7/2  +  2xy  =  455.  1  y2  +  2x2  =  99. 


,  , 

=  2x  +  5y.  1  2x  +  y  =  3. 

f  x(3x  +  y)  =  j/(r/  +  3),  f  3x(x  -  4)  =  y  -  5, 

1  (3x  -  y)(x  -  2y  +  3)  =  0.  1  2x  +  y  =  30. 

2  +  4j/2  =  48, 


10      \  20 

V  +  y2  =  58  U  +  3(x  +  l)  =0. 

f  x2  +  xy  +  y2  =  7,  f  2x(2x  -  3)  =  184, 

1  x2  -  xy  +  y2  =  19.  1  9y(2x  +  y)  =  -  135. 

f  x2  -  3xy  +  y2  =  1,  „.      f  x2  +  xy  +  y2  =  7, 

1  (x+y+2)(2x-2/  +  l)  =0,  1  y2  -  x2  =  5. 

'  x2  +  3xy  +  y2  -  4x  -  2y  -  1  =  0, 

0O« 


15x  +  4y  -  1  =  0. 
26'     I  x  +  2xy  +  y  -  17  =  0. 


CHAPTER  X 
VARIATION 

167.  Function  and  Variables.  One  of  the  most  common 
scientific  problems  is  to  investigate  the  causes  or  effects  of 
certain  changes.  The  change  or  variation  of  one  quantity  in 
the  problem  is  produced  or  caused  by  changes  in  other  variable 
quantities  and  is  said  to  depend  upon,  or  be  a  function  of  these 
variables.  Thus  the  growth  of  a  plant  depends  on  the  amount 
of  certain  constituents  in  the  soil,  upon  the  temperature  and 
humidity  of  the  soil  and  of  the  atmosphere,  upon  the  intensity 
of  the  light,  and  doubtless  upon  several  other  variables.  The 
volume  of  gas  contained  in  an  elastic  bag  depends  on  the  pressure 
and  the  temperature.  The  circumference  of  a  circle  depends 
only  on  the  radius. 

To  study  the  effect  of  any  one  variable  upon  a  function  of 
two  or  more  variables,  we  try  to  arrange  conditions  so  that 
all  the  other  variables  of  the  problem  shall  remain  constant, 
while  this  one  varies.  Thus  we  keep  the  temperature  of  a  gas 
constant  to  find  the  effect  on  the  volume  of  a  change  of  the 
pressure.  To  study  the  effect  of  carbonate  of  lime  on  the 
growth  of  alfalfa,  we  arrange  a  series  of  plats  of  soil  so  that 
they  shall  have  all  the  other  constituents  the  same,  and  all 
be  subject  to  the  same  conditions  of  light,  heat,  and  moisture, 
but  differ  from  plat  to  plat  by  known  amounts  of  pulverized 
limestone. 

The  precise  form  of  the  relation  between  a  function  and  its 
variables  is  often  very  complicated  and  difficult  or  impossible 
to  obtain.  Often,  the  best  that  can  be  done  is  to  record  the 
results  of  experiments  and  to  study  these  records  to  deduce 

218 


X,  §170]  VARIATION  219 

general  effects.  Such  results  are  called  empirical.  This  is 
especially  true  of  the  so-called  applications  of  science  to  the 
processes  of  nature. 

168.  Direct  Variation.  One  of  the  simplest  relations  that 
can  exist  between  two  variables  is  called  direct  variation. 
When  the  ratio  of  two  variables  is  constant,  each  is  said  to 
vary  directly  as  the  other. 

The  statement  that  y  varies  directly  as  x  or  simply  y  varies 
as  x,  is  written 


which  means  that  the  ratio  y/x  is  constant  and  implies  the 
equation 

y  =  kx, 

where  k  is  called  the  constant  of  variation. 

The  circumference  of  a  circle  varies  as  the  radius;  i.  e.,  C  °c  r, 
or  C  =  kr.  The  constant  of  variation  is  known  to  be  k  =  2ir 
—  6f ,  approximately. 

169.  Inverse  Variation.     When  the  product  of  two  variables 
is  constant,  each  is  said  to  vary  inversely  as  the  other.     If  y 
varies  inversely  as  x,  then 

xy  =  k,         or         y  =  k  I  -  )  , 

\  *  / 

whence  y  varies  directly  as  1/x,  the  reciprocal  of  x. 

EXAMPLE.  The  volume  v,  of  a  gas  kept  at  constant  temperature, 
varies  inversely  as  the  pressure  p;  i.  e. 

k  k 

pv  =  k,        or        v  =  -  ,         or         p  =  - . 
p  v 

170.  Joint  Variation.     When  a  function  z  depends  upon  two 
variables  x  and  y,  in  such  a  manner  that  z  varies  as  the  product 
xy,  i.  e.,  z  =  kxy,  then  z  is  said  to  vary  jointly  as  x  and  y.     Thus, 
the  area  of  a  rectangle  varies  jointly  as  the  length  and  the 


220 


MATHEMATICS 


[IX,  §  170 


breadth.  This  definition  may  be  extended  to  functions  of  three 
or  more  variables.  A  function  /,  depending  upon  several  vari- 
ables x,  y,  •  -  • ,  z,  is  said  to  vary  jointly  as  x  and  y,  •  •  • ,  and  z, 
when  it  varies  as  their  product,  i.  e.,  /  =  kx-y  •  -  -  z.  Thus, 
simple  interest  varies  jointly  as  the  principal,  and  the  rate, 
and  the  time. 

It  is  evident  that  if  one  variable  z  depends  on  two  other 
variables  x  and  y,  and  if  z  varies  as  x  when  y  is  constant,  and  z 
varies  as  y  when  x  is  constant,  then  z  varies  jointly  as  x  and  y 
when  x  and  y  vary  simultaneously.  Thus,  the  area  of  a  triangle 
varies  as  the  altitude  when  the  base  is  constant  and  varies  as 
the  base  when  the  altitude  is  constant;  therefore  the  area  varies 
jointly  as  the  base  and  the  altitude. 

This  principle  is  readily  extended  to  functions  of  three  or 
more  variables.  Thus,  simple  interest  varies  as  the  principal 
when  rate  and  time  are  constant,  as  the  rate  when  principal 
and  time  are  constant,  and  as  the  time  when  principal  and  rate 
are  constant;  therefore  simple  interest  varies  jointly  as  the  prin- 
cipal, the  rate,  and  the  time. 

171.  Graphic  Representation.  When  y  varies  directly  as 
x,  the  graph  of  the  relation,  y  =  kx,  which  connects  them  is 


1 

X 

x* 

X 

X 

S 

X 

^ 

^ 

X 

x 

f 

n-"*1 

•^ 

^ 

? 

o 

f* 

x 

.Y 

^ 

1 

2 

? 

J 

5 

1 

1 

FIG.  122 

a  straight  line  through  the  origin  whose  slope  is  k.     The  position 
of  this  line  is  fixed  and  the  value  of  k  can  be  determined  if  we 


X,  §  171] 


VARIATION 


221 


know  one  other  point  on  the  line,  i.  e.,  one  pair  of  simultaneous 
values  of  x  and  y;  and  values  of  y  corresponding  to  any  given 
values  of  x,  can  be  read  directly  from  the  graph.  Then  k  is  the 
difference  of  two  values  of  y  divided  by  the  difference  of  the 
corresponding  values  of  x  (§  58). 

EXAMPLE.     Given  that  y  varies  as  x  and  that  y  =  1  when  x  =  2. 

Plotting  the  point  (2,  1)  and  drawing  the  line  OP  we  have  the  graph 
of  the  relation  between  x  and  y.  From  this  we  read  off  y  =  |  when 
x  =  1,  y  =  3i  when  x  =  6|,  etc.  Fig.  122. 

When  y  varies  inversely  as  x,  the  graph  of  their  relation 
xy  =  k  is  a  rectangular  hyperbola  asymptotic  to  the  x  and  y 
axes.  Here  again  one  point  is  sufficient  to  determine  k  and  fix 
the  curve. 

EXAMPLE.  Given  that  volume  v,  varies  inversely  as  pressure  p, 
and  that  v  =  12  when  p  =  3. 

Then  pv  —  k,  3-12  =  k,  pv  =  36.     The  graph  of  this  is  shown  in 


10 


10 


FIG.  123 

Fig.  123  for  positive  values  of  p  and  v.     From  this  we  can  read  off 
t;  =  6  when  p  =  6,  v  =  4  when  p  =  9,  etc.* 

*  When  z  varies  jointly  as  x  and  y,  the  graph  of  the  relation  z  =  kxy,  in  three 
dimensions,  is  a  surface  called  a  hyperbolic  paraboloid  with  which  the  student  is  not 
yet  familiar. 


24 


222  MATHEMATICS  [X,  §  172 

172.  Determination  of  the  Constant.  By  substituting  in 
an  equation  of  variation  a  set  of  simultaneous  values  of  the 
variables,  the  constant  of  variation  can  be  determined. 

EXAMPLE  1.  Given,  y  varies  as  x  and  y  =  8  when  x  =  10.  We 
may  write  y  =  kx,  as  in  §  168.  Substituting  x  =  10  and  y  =  8,  we 
find  8  =  ft- 10.  From  this  equation,  we  can  find  k  by  dividing  both 
sides  by  10.  This  gives  k  =  4/5.  Hence  we  have  y  =  (4/5)x. 

From  this  equation,  the  value  of  y  corresponding  to  any  given 
value  of  x  can  be  found.  Thus,  y  =  If  when  x  =  2. 

EXAMPLE  2.     A  light  is  24  inches  above  the  cen- 
ter of  a  table.     The  illumination  I  at  any  point  P  of 
the  surface  of  the  table  varies  directly  as  the  cosine 
of  the  angle  of  incidence,  i,  of  the  ray  LP,  and  also 
•p ^24  inversely  as  the  square  of  the  distance  LP  =  x  to  the 

light.     If  the  illumination  at  C  is  10,  what  is  it  at 
any  point  P  of  a  circle  of  radius  18  inches  about  C? 
SOLUTION.     The  illumination  7  at  any  point  is 

T  _  ,  cos  i 
:^>?~' 

but  x  =  24  sec  i  and  therefore 

i* 

-r  A/  .      . 

7  =mco#t. 

Since  7  =  10  when  i  =  0,  k  =  5760,  and  hence 

7  =  10  cos'  i. 
Now  when  CP  =  18,  cos  i  =  4/5,  and  7  at  P  is  equal  to  5.12 

EXERCISES 

1.  Write  equations  equivalent  to  each  of  the  following  statements; 
determine  the  constant  of  variation  and  construct  the  graph. 

(a)  y  varies  as  x;  y  =  7  when  x  =  3. 

(6)  y  is  proportional  to  x;  y  =  3  when  x  =  2|. 

(c)  y  varies  inversely  as  x;  y  =  1J  when  x  =  1-|. 

(d)  v  varies  inversely  as  p]  v  =3  when  p  —  2. 

2.  Write  equations  equivalent  to  each  of  the  following  statements 
and  find  the  value  asked  for  in  each  case. 


X,  §  172]  VARIATION  223 

(a)  y  varies  as  x2 ;  y  =  81  when  x  =  3 ;  find  y  when  x  =  51. 

(b)  y  varies  as  sin  x ;   y  —  2  when  x  =  30° ;   find  y  when  x  =  150°. 

(c)  u  varies  inversely  as  v ;  u  =  8  when  v  =  2 ;  find  u  when  v  =  6. 

(d)  z  varies  jointly  as  x  and  y;z  =  Q  when  x  =  2,  y  =  7 ;  find  2  when 

x  =  4,  y  =  6. 

(e)  y  varies  directly  as  r  and  inversely  as  s ;  y  =  16  when  r  =  10,  s  =  8 ; 

find  y  when  r  =  7,  s  =  12. 
(/)    M  varies  jointly  as  x,  and  y2,  and  z"1 ;    tt  =  6  when  x  =  2,  y  =  3, 

2  =  4;   find  w  when  x  =  10,  ?/  =  15,  2  =  25. 
(0)  2  varies  directly  as  x  and  inversely  as  y2 ;  z  =  2  when  x  =  17,  y  =  3 ; 

find  x  when  2  =  6,  y  =  4. 

3.  Express  each  of  the  following  by  means  of  an  equation. 

(a)  The  volume  of  a  cone  varies  directly  as  the  height  when  the  radius 

of  the  base  is  constant. 

(b)  The  volume  of  a  cone  varies  directly  as  the  square  of  the  radius  of 

the  base  when  the  height  is  constant. 

(c)  The  number  of  calories  of  heat  produced  when  a  moving  body  is 

stopped  varies  jointly  as  the  mass  and  the  square  of  the  velocity. 

(d)  The  squares  of  the  periods  of  the  planets  vary  directly  as  the  cubes 

of  their  mean  distances  from  the  sun. 

4.  With  the  statement  of  Ex.  3  (c)  find  the  heat  generated  by  a  mass 
of  8  kilograms  striking  the  sun  with  a  velocity  of  500  miles  per  second 
if  a  body  weighing  one  kilogram  and  moving  with  a  velocity  of  380 
miles  per  second  on  striking  the  sun  produces  45,000,000  calories  of  heat. 

5.  The  simple  interest  due  on  P  dollars  varies  jointly  as  the  amount 
P,  the  rate,  and  the  time.     If  $1000  yields  $30  interest  in  six  months 
find  the  interest  on  $1200  for  eight  months  at  7%. 

6.  The  amount  of  heat  received  by  a  given  planet  varies  inversely 
as  the  square  of  its  distance  from  the  sun  and  directly  as  the  square  of  its 
radius. 

(a)  What  is  the  effect  of  doubling  the  distance? 

(b)  Mercury  has  a  diameter  of  3000  miles  and  is  36  million  miles 
from  the  Sun.  The  Earth  has  a  diameter  of  8000  miles  and  is  93  million 
miles  from  the  Sun.     Compare  the  amounts  of  heat  they  receive. 

7.  With  the  statement  in  Ex.  3(d),  taking  the  distance  of  the  Earth 
from  the  Sun  as  the  unit  and  the  period  of  the  Earth  as  the  unit  of  time, 


224  MATHEMATICS  [X,  §  172 

find  the  period  of  Neptune  whose  distance  from  the  Sun  is  known  to  be 
30  units.  Ans.    165  yrs. 

8.  The  amount  of  heat  received  on  a  surface  of  given  size  varies  in- 
versely as  the  distance  from  the  source.     One  body  is  twice  as  far  as 
another  from  the  source.     Compare  the  amounts  of  heat  received. 

9.  The  resistance  offered  to  a  rifle  bullet  varies  directly  as  the  square 
of  the  velocity.     Discuss  the  effect  of  doubling  the  velocity. 

10.  The  maximum  load  P  that  a  rectangular  beam  supported  at  one 
end  will  hold  without  breaking  varies  directly  as  the  breadth,  the  square 
of  the   depth  and   inversely  as  the  length.      A  beam  4"  X  2"  X  10' 
supports  300  pounds.     What  load  will  the  same  beam  support  when 
placed  on  edge? 

11.  The  deflection  y  in  a  rectangular  beam  supported  at  the  ends  and 
loaded  in  the  middle  varies  directly  as  the  cube  of  the  length,  inversely 
as  the  breadth,  and  inversely  as  the  cube  of  the  depth.     A  beam  6  inches 
wide,  8  inches  deep,  15  feet  long,  supporting  1000  Ibs.,  has  a  deflection 
of  \  inch  at  the  middle.     Find  the  deflection  in  a  beam  4  inches  wide, 
4  inches  deep,  10  feet  long,  supporting  800  Ibs. 

12.  With  the  data  of  Ex.  10,  find  the  load  which  a  beam  4  inches  wide, 
6  inches  deep,  and  16  feet  long  will  support. 

13.  With  the  data  of  Ex.  10,  find  the  longest  beam  4  inches  wide  and 
4  inches  deep  which  will  support  100  Ibs. 

14.  With  the  data  of  Ex.  10,  find  the  least  depth  of  a  beam  12  feet 
long  and  4  inches  wide  that  will  support  400  Ibs. 

15.  With  the  data  of  Ex.  10,  find  the  least  breadth  of  a  beam  12  feet 
long  and  4  inches  deep  that  will  support  500  Ibs. 

16.  Evaporation  from  a  surface  varies  directly  as  its  area. 

(a)  Of  two  square  vats  the  side  of  one  is  10  times  that  of  the  other. 

What  is  the  ratio  of  evaporation? 
(5)  Of  two  circular  vats  one  evaporates  10  times  as  fast  as  the  other. 

Compare  their  radii. 

17.  The  distance  traversed  by  a  falling  body  varies  directly  as  the 
square  of  the  time.     If  a  body  falls  144  feet  in  3  seconds,  how  far  will 
it  fall  in  5  seconds? 

18.  The  area  of  a  triangle  varies  jointly  as  the  length  of  the  base  b 
and  the  altitude  a.     Write  the  law  if  the  area  is  12  square  inches  when 
a  =  6  inches  and  b  =  4  inches. 


X,  §172]  VARIATION  225 

19.  Similar  figures  vary  in  area  as  the  squares  of  their  like  dimensions. 
A  new  grindstone  is  48  inches  in  diameter.     How  large  is  it  in  diameter 
when  one-fourth  of  it  is  ground  away? 

20.  A  circular  silo  has  a  diameter  of  a  feet.     What  must  be  the 
diameter  of  a  circular  silo  of  the  same  height  to  hold  4  times  as  much? 

21.  What  is  the  effect  on  the  area  of  a  regular  hexagon  if  the  length 
of  each  side  of  the  hexagon  is  doubled. 

22.  Similar  solids  vary  in  volume  as  the  cubes  of  their  like  dimen- 
sions.    A  water  pail  that  is  10  inches  across  the  top  holds  12  quarts. 
Find  the  volume  of  a  similar  pail  that  is  12  inches  across  the  top. 

23.  Using  the  rectangular  pack,  432  apples  2  inches  in  diameter  can 
be  put  in  a  box  12  X  12  X  24.     How  many  3  inch  apples  can  be  packed 
in  the  same  box?     How  many  4  inch  apples?  Ans.  128;  54. 

24.  If  a  lever  with  a  weight  at  each  end  is  balanced  on  a  fulcrum, 
the  distances  of  the  two  weights  from  the  fulcrum  are  inversely  propor- 
tional to  the  weights.     If  2  men  of  weights  160  Ibs.  and  190  Ibs.  respect- 
ively are  balanced  on  the  ends  of  a  10  foot  stick,  what  is  the  length 
from  the  fulcrum  to  each  end?  Ans.  4^  ft.;  5f  ft. 

25.  A  wire  rope  1  inch  in  diameter  will  lift  10,000  Ibs.     What  will 
one  f  inches  in  diameter  lift?  Ans.  1,406  Ibs. 

26.  Two  persons  of  the  same  build  are  similar  in  shape;  their  weights 
should  vary  as  the  cube  of  their  heights.     A  man  5|  ft.  tall  weighs 
150  Ibs.     Find  the  weight  of  a  man  of  the  same  build  and  6  feet  tall. 

Ans.  194.74  Ibs. 

27.  A  man  5  ft.  5  in.  tall  weighs  140  Ibs.,  and  one  6  ft.  2  in.  tall  weighs 
216  Ibs.     Which  is  of  the  stouter  build? 

28.  The  size  of  a  stone  carried  by  a  swiftly  flowing  stream  varies  as 
the  6th  power  of  the  speed  of  the  water.     If  the  speed  of  a  stream  is 
doubled,  what  effect  does  it  have  on  its  carrying  power?    What  effect 
if  trebled? 


16 


CHAPTER  XI 

EMPIRICAL  EQUATIONS 

173.  Empirical  Formulas.  In  practice,  the  relations  be- 
tween quantities  are  usually  not  known  in  advance,  but  are  to 
be  found,  if  possible,  from  pairs  of  numerical  values  of  the 
quantities  discovered  from  experiment. 

In  order  to  determine  the  relation  between  these  quantities 
it  is  useful  to  first  plot  the  corresponding  pairs  of  values  upon 
cross-section  paper,  and  draw  a  smooth  curve  through  the 
plotted  points.  If  the  curve  so  drawn  resembles  closely  one 
of  the  following  types  of  curves: 

(1)  y  =  ax  +  b  (straight  line), 

(2)  y  =  a  +  bx  +  ex2  (parabola), 

(3)  x  =  a  +  by  +  c?/2  (parabola), 

(4)  y  =  kxn  (parabolic  in  form), 

(5)  xy  =  c  (hyperbola), 

(6)  y  =  c!0kx  (exponential  curve), 

we  assume  that  the  relation  connecting  the  quantities  is  the 
corresponding  equation  of  the  above  set  and  it  remains  to 
determine  the  constants  of  the  equation. 

If  the  plotted  data  does  not  fit  any  of  the  type  curves  men- 
tioned above,  a  general  method  of  procedure  is  to  assume  an 
equation  of  the  type 

(7)  y  =  OQ  +  a\x  +  a2z2  +  •  •  •  +  anxn     (nth  degree  curve) . 

The  coefficients  OQ,  a\,  a%,  •  •  • ,  an  can  be  found  from  any  n  +  1 
pairs  of  values  of  x  and  y. 

Since  the  measurements  made  in  any  experiment  are  liable 

226 


XI,  §  174] 


EMPIRICAL  EQUATIONS 


227 


to  be  in  error,  errors  will  occur  in  the  computed  values  of  the 
coefficients.  The  curve  represented  by  the  final  equation  will 
not  in  general  pass  through  the  points  representing  the  ob- 
served data.  Some  of  these  points  will  be  on  one  side  and 
some  on  the  other.  All  will  be  near  the  curve. 

174.  Computation  of  the  Coefficients  in  the  Assumed 
Formula.  In  case  the  plotted  points  appear  to  be  upon  a 
straight  line,  a  parabola,  or  a  curve  of  the  nth  degree,  the 
corresponding  equation  is  assumed  and  we  proceed  to  determine 
the  coefficients  by  a  method  which  is  illustrated  in  the  following 
example. 

EXAMPLE  1.     Let  the  observed  values  of  x  and  y  be 


X  

43 

85 

127 

169 

V  .  . 

17 

33 

49 

65 

Plotting  this  data,  the  points  will  be  seen  to  lie  roughly  on  a  straight 
line.     Hence  we  assume  a  relation  of  the  form 

y  =  ax  +  b. 


y 


•10 


o 


40 


120 


160 


80 

FIG.  125 

In  this  equation  replace  x  and  y  by  their  observed  values.     In  this  way 


228  MATHEMATICS  [XI,  §  174 

we  obtain  four  equations  connecting  a  and  b : 
43a  +  b  =  17, 
85a  +  b  =  33, 
127a  +  b  =  49, 
169a  +  b  =  65. 

Two  equations  are  necessary  and  sufficient  for  the  determination  of  the 
two  unknowns  a  and  b.  In  general  if  we  have  more  equations  than 
unknowns  the  equations  are  not  consistent.  That  is,  the  values  of 
a  and  b  as  determined  from  the  first  two  equations  are  not  the  same 
as  those  obtained  from  the  last  two,  or  from  the  second  and  third,  etc. 
Our  problem  then  is  to  derive  from  the  given  set  two  equations  such 
that  the  values  of  a  and  b  obtained  therefrom  when  used  as  coefficients 
in  the  assumed  equation  will  give  us  a  straight  line  which  fits  closely 
the  points  plotted  from  the  observed  data.  There  are  in  common  use  a 
number  of  ways  of  doing  this. 

FIRST  METHOD.     Multiply  each  equation  in  turn  by  the  coefficient 
of  a  in  that  equation  and  add.     This  gives  one  equation  containing  a 
and  b.     Multiply  each  equation  in  turn  by  the  coefficient  of  b  in  that 
equation  and  add.     This  gives  a  second  equation  containing  a  and  b. 
Using  the  data  in  (8)  above  we  find  in  this  way  the  following  equations : 
53764a  +  4246  =  20744, 
424a  +      46  =       164. 
The  solution  of  these  equations  for  a  and  6  gives 

(10)  a  =  0.39,  6  =  -  0.34 
Substituting  these  values  of  a  and  6  in  the  assumed  equation,  we  find 

(11)  y  =  0.39*  -  0.34. 

SECOND  METHOD.  When  on  plotting  it  is  clear  that  a  straight  line 
is  the  best  fitting  curve,  draw  a  straight  line  among  the  points  so  that 
about  half  are  above  and  half  below.  The  y  coordinate  of  the  inter- 
section of  this  line  with  the  y-axis  can  then  be  read  directly  from  the 
graph  and  gives  the  value  of  6  in  the  equation  y  =  ax  +  b.  Measure 
the  angle  a  that  this  line  makes  with  the  z-axis  and  then  a  =  tan  a. 

In  ca*se  different  scales  are  used  on  the  two  axes  select  two  points 
(xi,  yi)  (x2,  2/2)  on  the  line,  then 

(12)  0=W2-H-Wl. 

Xt  —  Xi 


XI,  §-174] 


EMPIRICAL  EQUATIONS 


229 


THIRD  METHOD.  Suppose  the  best  fitting  curve  is  a  straight  line, 
i.e.  that  the  equation  should  be  of  the  form 

y  =  ax  +  b. 

Use  for  a  and  b  the  values  obtained  on  solving  the  first  and  last  of  equa- 
tions (8).  The  straight  line  so  found  actually  passes  through  the  first 
and  last  points. 

If  the  points  are  so  distributed  that  one  of  the  forms  (2)  or  (3)  §  173 
should  be  used,  proceed  to  find  a,  b,  and  c  by  using  the  first,  middle,  and 
last  points  only. 

If  an  equation  of  degree  n  [(7),  §  173],  i.e.  an  equation  of  the  form 

y  =  QO  4-  o,\x  +  a2x2  +  •••  +  anxn 

should  be  assumed,  use  n  +  1  points  evenly  distributed  along  the 
curve.  This  method  gives  us  always  the  same  number  of  equations  as 
there  are  unknown  coefficients  to  be  determined. 

FOURTH  METHOD.  If  it  is  known  that  the  curve  is  a  straight  line 
through  the  origin  then 

y  =  kx. 

Substitute  the  observed  pairs  of  values  of  x  and  y  in  this  equation, 
add  the  resulting  equations  and  solve  for  k.  See  §  172. 

EXERCISES 

1.  In  the  following  example  a  series  of  observed  values  of  y  and  x 
are  given.  The  variables  are  known  to  be  connected  by  a  relation 

of  the  form 

y  =  ax  +  b. 

Ans.   a  =  0.498,  b  =  0.96 


Find  a  and  b. 


11.  . 

6 

10.8 

16.1 

20.6 

26 

X  

10 

20 

30 

40 

50 

2.    The  following  table  gives  the  density  8  of  liquid  ammonia  at  vari- 
ous degrees  centigrade.     Find  a  relation  of  the  form 

5  =  at  +  b. 
i.e.  determine  the  values  of  a  and  b. 


t  

0 

5 

10 

15 

s  

.6364 

.6298 

.6230 

.6160 

Ans.   d  =  0.6364  -  0.0014  t 


230 


MATHEMATICS 


[XI,  §  174 


3.   The  following  table  gives  the  specific  heat  s  of  hot  liquid  ammonia 
at  various  degrees  Fahrenheit.     Find  a  relation  of  the  form  s  =  at  +  b. 


t  .  . 

5 

10 

15 

20 

25 

s 

1  090 

1  084 

1  078 

1.072 

1  066 

Ans.    s  =  1.096  -  0.0012i 

4.  In  an  experiment  to  determine  the  coefficient  of  friction  between 
two  surfaces  (oak)  the  following  values  of  F  were  required  to  give  steady 
motion  to  a  load  W.  Plot  F  and  W  on  squared  paper,  and  find  M  where 
M  =  F/W.  [CASTLE]  Ans.  M  =  3.302 


F 

5 

10 

15 

20 

25 

30 

35 

40 

W 

2 

3 

6} 

74 

10V 

111 

2 

2 

5.  In  the  following  examples  a  series  of  values  of  x  and  y  are  given. 
In  each  case  the  variables  are  connected  by  an  equation  of  the  form 
y  =  ax  +  b.  Find  a  and  b. 


(a) 


(b) 


(c) 


(d) 

Ans.   a  =  0.33,  b  =  0.7 

In  the  two  following  sets  of  data  plot  the  values  of  E  (Electromotive 
force)  and  R  (Resistance),  and  determine  an  equation  of  the  form 

E  =  aR  +  b. 


11.  . 

5 

7.8 

11.1 

14.2 

17 

X  

9 

18 

27 

36 

45 

Ans. 

a  =  0.337 

,  b  =  1.9 

y  

2 

3.1 

4 

5.2 

6.2 

x  

4 

8 

12 

16 

20 

Ans.  a 

=  0.2625, 

6  =  0.95 

y  .  . 

5 

6.1 

8.2 

10 

12.1 

X  .  . 

1 

2 

3 

4 

5 

Ans. 

a  =  1.81, 

b  =  2.85 

y  

4 

7 

11 

14 

17 

X  

10 

20 

30 

40 

50 

XI,  §  174] 


EMPIRICAL  EQUATIONS 


231 


E.  . 

0 

5 

1 

1  5 

2 

9  5 

3 

3 

5 

4 

45 

5 

R  

7 

5 

18 

28 

38 

49 

t 

>9 

f 

58 

8(1 

90 

100 

E.. 
R.... 

1 

3 
4 

4 

( 

[.5 

28 

6 

42 

7.75 
56 

9.5 
70 

1] 

& 

I 
I 

IS 

g 

.5 

8 

13.5 
112 

15 
126 

6.  A  wire  under  tension  is  found  by  experiment  to  stretch  an  amount 
I,  in  thousandths  of  an  inch,  under  a  tension  T,  in  pounds,  as  follows: 


T.  . 

10 

15 

20 

25 

30 

1  

8 

12.5 

155 

20 

23 

Find  a  relation  of  the  form  I  =  kT  (Hooke's  law)  which  best  represents 
these  results. 

7.  In  an  experiment  with  a  Weston  differential  pulley  block,  the 
effort  E,  in  pounds,  required  to  raise  a  load  W,  in  pounds,  was  found 
to  be  as  follows: 


W.  . 

10 

90 

30 

40 

50 

60 

70 

80 

90 

100 

E  

•H 

4f 

61 

74 

9 

10| 

m 

13f 

15 

16i 

Find  a  relation  of  the  form  E  =  aW  +  b. 

8.  If  6  denotes  the  melting  point  (Centigrade)  of  an  alloy  of  lead 
and  zinc  containing  x  per  cent,  of  lead,  it  is  found  that 


x  

40 

50 

60 

70 

80 

90 

e  

186 

205 

226 

250 

276 

304 

Find  a  relation  of  the  form  0  =  a  +  bx  +  ex2. 

9.  The  readings  of  a  standard  gas-meter  S  and  those  of  a  meter  T 
being  tested  on  the  same  pipe  line  were  found  to  be 


S  . 

3,000 

3  510 

4022 

4  533 

T  

0 

500 

1,000 

1,500 

Find  a  formula  of  the  type  T  =  aS  +  b  which  best  represents  these 
data.     What  is  the  meaning  of  a?  of  fe? 


232 


MATHEMATICS 


[XI,  §174 


10.  An  alloy  of  tin  and  lead  containing  x  per  cent,  of  lead  melts  at 
the  temperature  0  (Fahrenheit)  given  by  the  values 


X  

25 

50 

75 

e  

482 

370 

356 

Determine  a  formula  of  the  type  6  =  a  +  bx  +  ex2. 

11.  A  restaurant  keeper  finds  that  if  he  has  G  guests  a  day  his  total 
daily  expenditure  is  E  dollars,  and  his  total  daily  receipts  are  R  dollars. 
The  following  numbers  are  averages,  obtained  from  the  books 


G.  . 

210 

270 

320 

360 

E  

16.7 

19.4 

21.6 

23.4 

R  

15.8 

21.2 

26.4 

29.8 

Find  the  simple  algebraic  laws  which  seem  to  connect  E  and  R  with 
G.  [R  =  mG;  E  =  aG  +  b.]  What  are  the  meanings  of  m,  a,  and  fe? 
Below  what  value  of  G  does  the  business  cease  to  be  profitable? 

12.  The  following  statistics  (taken  from  Bulletin  110,  part  1  of  the 
Bureau  of  Animal  Industry,  U.  S.  Dept.  of  Agriculture)  give  the  changes 
in  average  egg  production  between  1899  and  1907: 


Year. 

Birds 
Competing 
per  Year. 

Eggs  Laid. 

Actual 
Average 
Production. 

Added  to 
Actual 
Average. 

Modified 
Average 
Due  to 
Abnormal 
Conditions. 

1899-1900  

70 

9,545 

136.36 

0 

136.36 

1900-    01    

85 

12,192 

143.44 

0 

143.44 

01-    02  

48 

7,468 

155.58 

0 

155.58 

2-      3  

147 

19,906 

135.42 

23.73 

159.15 

3-      4  

254 

29,947 

117.90 

11.24 

129.14 

4-      5  

283 

37,943 

134.07 

0 

134.07 

5-      6  

178 

24,827 

140.14 

13.95 

154.09 

6-      7.. 

187 

21.175 

113.24 

29.53 

142.77 

With  the  actual  and  modified  averages  in  hand  we  may  inquire: 
what  has  been  the  general  trend  of  the  mean  annual  egg  production 
during  the  period  covered  by  the  investigation?  The  clearest  answer 
to  this  question  may  be  obtained  by  plotting  the  figures  in  the  fourth 
and  sixth  columns  of  the  above  table,  and  then  striking  through  each 


XI,  §175] 


EMPIRICAL  EQUATIONS 


233 


of  the  two  zigzag  lines  so  obtained  the  best  fitting  straight  line,  as 
determined  by  the  method  of  least  squares.  The  equations  of  the 
two  straight  lines  are  as  follows: 


actual  averages: 
modified  averages: 


y  =  148.48  -  3.10z, 
y  =  144.13  +  0.043z. 


In  these  equations  y  represents  the  mean  annual  egg  production  and 
x  the  year.  The  origin  for  x  is  at  1898-99.  Verify  these  two  equations. 

13.  The  following  table,  taken  from  the  same  bulletin,  gives  the 
percentage  of  the  flocks  laying  (a)  less  than  45  eggs,  and  (6)  195  or  more 
eggs  in  a  year. 


Annual  Egg  Production. 

1899- 
1900. 

1900- 
•81. 

•01-'02. 

•02- 
'03. 

•03- 
'04. 

•04-'05. 

'05- 
•06. 

•06- 
•07. 

Less  than  45  in  %  .  . 

4.29 

1.18 

0 

1.36 

6.70 

7.07 

0.56 

4.81 

195  or  more  in  %  ... 

4.29 

10.60 

18.75 

6.12 

0.79 

12.71 

5.06 

0 

Plot  this  data,  using  years  for  abscissa  and  percentages  for  ordinates, 
making  two  curves  and  find  by  the  method  of  least  squares  the  best 
fitting  lines. 

Poor  layers:     y  =     1.795  +  0.3225x. 

Good  layers:     y  =  11.639  -  0.966x. 

Interpret  the  sign  of  the  coefficient  of  x  in  each  equation,  and  give  the 
meaning  of  the  constant  term  in  each  equation. 

175.  Substitution.  If  on  plotting  the  given  values  of  x 
and  y  the  plotted  points  are  seen  to  be  approximately  on  a 
branch  of  a  rectangular  hyperbola  with  vertical  and  horizontal 
asymptotes  we  assume  a  relation  of  the  form 

(14)  (x  -  a)(y  -b)  =c, 

where  (a,  6)  are  the  coordinates  of  the  intersection  of  the  asymp- 
totes, and  proceed  to  determine  a,  b,  and  c. 

In  many  of  the  cases  in  which  this  form  appears  both  a  and  6 
are  zero  and  the  equation  (14)  becomes 

y  =  c/x. 


234  MATHEMATICS  [XI,  §175 

In  some  cases  a  is  zero  and  equation  (14)  becomes 
y  =  b  +  c/x. 

There  are  many  curves  which  resemble  closely  the  curve 
given  by  equation  (14),  but  whose  equation  is  somewhat  differ- 
ent. In  order  to  determine  whether  (14)  is  the  best  equation 
to  represent  the  plotted  data,  obtain  from  the  figure  an  approxi- 
mate value  of  a.  In  many  cases  a  =  0.  Make  the  substitution 
l/(x  —  a)  =  u  and  plot  the  new  points  (u,  y).  If  these  are 
approximately  upon  a  straight  line  then 

y  —  b  +  cu* 

and  equation  (14),  in  one  of  its  forms,  is  the  proper  relation  to 
assume. 

If  on  plotting  the  observed  values  of  x  and  y  the  plotted  points 
appear  to  be  on  a  parabola  with  axis  parallel  to  one  of  the  axes 
and  vertex  on  that  axis  then  call  that  axis  the  ?/-axis  and  assume 
(15)  y  =  a  +  bx2.  ' 

The  determination  of  the  coefficients  a  and  b  can  be  reduced 
to  that  of  finding  the  coefficients  in  the  linear  form 

y  =  a  +  bu, 

where  u  =  xz.  As  a  check  that  (15)  is  the  correct  form  to  as- 
sume plot  pairs  of  values  of  u  and  y.  If  these  points  appear  to 
be  on  a  straight  line  then  equation  (15)  is  the  correct  form  to 
assume. 

EXAMPLE.  The  distance  s,  in  feet,  passed  over  by  a  falling  body  in  t 
seconds  is  found  by  experiment  to  be 


s             

0 
0 

5 
.5 

16 
1 

35 
1.5 

65 

2 

t             

Find  a  law  connecting  s  and  t. 

*This  is  sometimes  called  the  reciprocal  curve. 


XI,  §175] 


EMPIRICAL  EQUATIONS 


235 


Upon  plotting  this  data,  the  points  are  seen  to  fall  on  a  parabola  with 
vertex  upward  and  at  the  origin.  This  suggests  that  we  assume  the 
relation  of  the  form 

s  =  aP. 

As  a  check  on  this  assumption  we  plot  the  points  («,  s)  given  in  the 
following  table: 


5 
.5 

.25 


16 
1 
1 


35 
1.5 
2.25 


65 
2 
4 


These  points  are  approximately 
upon  a  straight  line  s  =  au.  The  de- 
termination of  a  by  the  method  of  least 
squares  gives  a  =  16.9,  whence 

s  =  16.9<2. 
EXERCISES 


10 


In 


1.  The  following  data  on  the  relation 
of  temperature  to  insect  life  gives  the 
number  of  days  at  a  given  temperature 
to  complete  a  given  stage  of  develop- 
ment and  is  taken  from  Technical  Bul- 
letin, No.  7,  Dec.  1913  of  the  New 
Hampshire  College  Ag.  Exp.  Station, 
each  case  the  plotted  points  are  on  a  curve  of  the  type 


FIG.  126 


y  —  b  =  c/x  (x  =  days,  y  =  temperature). 

The  term  developmental  zero  is  used  to  designate  that  point  at 
which  an  insect  may  be  kept,  theoretically  at  least,  without  change  for 
an  indefinite  period.  The  developmental  zero  for  the  insect  and  stage 
approximates  the  point  where  the  reciprocal  curve  (calculated  from  the 
time  factor)  intersects  the  temperature  axis.  (6  =  developmental 
zero.) 

For  each  set,  plot  the  data,  and  the  reciprocal  curve;  find  the 
developmental  zero,  and  obtain  an  equation  of  the  form  y  —  b  =  c/x 
connecting  the  data. 


236  MATHEMATICS  [XI,  §175 

(a)  Malacosoma  americana,  pupal  stage.     Developmental  zero  =  11°C. 


y    . 

32.4 

32 

26.1 

20 

16 

X  

9.7 

10 

13.2 

22.5 

54 

(b)   Tenebrio  molitor,  incubation  of  eggs.     Developmental  zero  =  9.5°  C. 


31.1 

26.6 

21 

11  6 

X 

6 

7.4 

12.1 

57 

(c)   Leptinotarsa  decemlineaia,  incubation  of  eggs.     Developmental  zero 
=  6°. 


32.2 

26.7 

18.6 

X 

3 

3.9 

6.3 

(d)  Toxoptera  graminum,  birth  to  death.     Developmental  zero  =  5° 


V    • 

32.5 

26.5 

21 

15.5 

10 

x  

10 

12 

20 

30 

58 

(e)   Incubation  period  of  eggs  of  codling  moth.     Developmental  zero 
=  6°. 


y 

28 

25 

22 

18 

16 

15 

X     .      ... 

4.5 

6 

7 

9 

12 

16 

(/)  Toxoptera  graminum,  birth  to  maturity.     Developmental  zero  =  5°. 


11 

26'.5 

21 

15.5 

10 

x      

6.5 

9 

15 

32 

Ans.  y  —  5  = 


150  (nearly) 


2.  Determine  a  relation  of  the  form  y  =  a  +  fez2  that  best  represents 
the  values. 


X 

1 

2 

3 

4 

5 

6 

11.  • 

14.1 

25.2 

44.7 

71.4 

105.6 

147.9 

3.  The  pressure  p,  measured  in  centimeters  of  mercury,  and  the 
volume  v,  measured  in  cubic  centimeters,  of  a  gas  kept  at  constant 
temperature,  were  found  to  be  as  follows. 


XI,  §176] 


EMPIRICAL  EQUATIONS 


237 


V  

145 

155 

165 

178 

191 

V  •  • 

117.2 

109.4 

102.4 

95 

88.6 

Determine  a  relation  of  the  form  pv  =  k. 

4.  Find  a  formula  of  the  type  u  =  kv2  that  best  represents  the 
following  values. 


u  

3.9 

15.1 

34.5 

61.2 

95.5 

137.7 

187.4 

V  

1 

2 

3 

4 

5 

6 

7 

5.  If  a  body  slides  down  an  inclined  plane,  the  distance  «,  in  feet, 
that  it  moves  is  connected  with  the  time  t,  in  seconds,  after  it  starts 
by  an  equation  of  the  form  s  =  kP.  Find  the  best  value  of  k  con- 
sistent with  the  following  data. 


s  

2.6 

10.1 

23 

40.8 

63.7 

t  

1 

2 

3 

4 

5 

Am.  k  =  2.556. 

6.  Find  approximately  the  relation  between  s  and  t  from  the  fol- 
lowing data. 


s    

3.1 

13 

30.6 

50.1 

79.5 

116.4 

/ 

.5 

1 

1.5 

2 

2.5 

3 

176.  Logarithmic  Plotting.  In  case  the  plotted  points  (x,  y} 
appear  to  lie  on  one  of  the  parabolic  or  hyperbolic  curves  of  the 
family 

(16)  y  =  bxm 

there  is  a  distinct  advantage  in  taking  the  logarithm  (base  10) 
of  both  sides: 

(17)  log  y  =  m  log  x  +  log  6, 

and  then  substitute 


(18) 


X  for  log  x,    Y  for  log  y,    B  for  log  b 


238 


MATHEMATICS 


[XI,  §176 


so  that  the  equation  (17)  becomes, 
(19)  F  =  mX  +  5. 

If  the  values  of  x  and  y  are  tabulated  in  columns,  and  their 
logarithms  X  and  Y  are  looked  up  and  written  in  parallel 
columns  opposite,  then  the  points  (X,  Y)  should  lie  on  a  straight 
line  to  justify  the  assumption  of  equation  (16).  And  if  they  do 
lie  fairly  on  a  line,  its  slope  and  y-intercept  determine  the  constants 
m  and  b  of  equation  (16).  This  can  often  be  done  graphically 
from  the  drawing  with  sufficient  accuracy,  but  if  greater  ac- 
curacy is  required  they  can  be  determined  from  the  data  by 
least  squares. 


EXAMPLE. 


X. 

y- 

A'=  log  z. 

r=iogi/. 

2 
4 
8 
16 

6.000 
24.60 
70.80 
338.8 

0.3010 
0.6020 
0.9030 
1.2040 

0.7782 
1.3909 
1.8500 
2.5299 

FIG.  127 


and  these  values  in  equation  (16)  give, 

y  =  1.574X1-"* 


The  points  (X,  F)  lie  nearly  on 
a  line  BD,  Fig.  127.  Graphically, 
we  scale  off  from  the  figure, 

B  =  the  t/-intercept  OB  =  0.2, 

CD 

m  —  the  slope  =  -^ 

-DO 

-47-188 
-25-1'88 

By  least    squares,   putting   the 
data  into  equation  (19),  we  find 

B  =  0.1970  =  log  6; 

hence 

b  =  1.574, 

m  =  1.914, 


XI,  §177] 


EMPIRICAL  EQUATIONS 


239 


In  case  the  quantities  x  and  y  are  connected  by  a  relation  of 
the  form 

(20)  y  =  cWkx, 

it  is  advantageous  to  compute  Y  =  log  y  and  plot  x  and  Y. 
If  these  new  values  when  plotted  appear  to  be  on  a  straight  line 
we  write 

(21)  F  =  kx  +  log  c 

and  determine  k  and  log  c  by  the  method  of  least  squares. 

177.  Logarithmic  Paper.  Paper,  called  logarithmic  paper, 
may  be  bought  that  is  ruled  in  lines  whose  distances,  horizontally 
and  vertically,  from  a  point  0  are  proportional  to  the  logarithms 
of  the  numbers  1,  2,  3,  etc. 

Such  paper  may  be  used  instead  of  actually  looking  up  the 
logarithms  in  a  table.  For  if  the  given  values  be  plotted  on  this 
new  paper,  the  resulting  "figure  is  identically  the  same  as  that 
obtained  by  plotting  the  logarithms  of  the  given  values  on  ordi- 
nary squared  paper. 

The  use  of  logarithmic  paper  is  however  not  essential;  it  is 
merely  convenient  when  one  has  a  large  number  of  problems 
of  this  type  to  solve. 

EXERCISES 

1.  A  strong  rubber  band  stretched  under  a  pull  of  p  kg.  shows  an 
elongation  of  E  cm.  The  following  values  were  found  in  an  experiment: 


p  

05 

1  0 

1.5 

?n 

?5 

30 

3  5 

40 

45 

5.0 

E  

0  1 

03 

Of> 

0.9 

1  3 

1  7 

?,?, 

2.7 

33 

3.9 

Find  a  relation  of  the  form  E  =  kpn.  Ans.  E  =  .Sp1-' 

2.  The  amount  of  water  A,  in  cu.  ft.,  that  will  flow  per  minute 
through  100  feet  of  pipe  of  diameter  d,  in  inches,  with  an  initial  pressure 
of  50  Ibs.  per  sq.  in.,  is  as  follows: 


d  

] 

1.5 

2 

3 

4 

6 

A  

4.88 

13.43 

27.50 

75.13 

152.51 

409.54 

Find  a  relation  of  the  form  A  =  kdn. 


Ans.  A  =  4.88eP-473 


240 


MATHEMATICS 


[XI,  §177 


3.  In  testing  a  gas  engine  corresponding  values  of  the  pressure  p, 
measured  in  Ibs.  per  sq.  ft.,  and  the  volume  v,  in  cubic  feet,  were  obtained 
as  follows: 


V  

7.14 

7.73 

859 

J>  •  • 

54.6 

50.7 

45.9 

Find  a  relation  of  the  form  p  =  kvn.  Ans.  p  =  387.6#~-938 

4.  Find  a  relation  between  p  and  v  from  the  following  data: 


v  

6.27 

534 

3  15 

V  •  • 

20.54 

25.79 

54.25 

Ans.  pvlM  =  273.5 

5.  The  intercollegiate  track  records  for  foot-races  are  as  follows, 
where  d  means  the  distance  run,  and  t  the  record  time: 


d 

100  yds. 

220  yds. 

440  yds. 

860  yds 

1  mi 

2  mi 

t 

0:094 

0:214 

0:48 

l-54f 

4-151 

9-241 

Find  a  relation  of  the  form  t  =  kdn.     What  should  be  the  record  time 
for  a  race  of  1,320  yds.? 

6.  In  each  of  the  following  sets  of  data  find  a  relation  of  the  form 
y  =  kxn  connecting  the  quantities. 


(a) 


(6) 


V  

1 

2 

3 

4 

5 

v  .  . 

137.4 

62.6 

39.6 

286 

226 

u  

12.9 

17.1 

23  1 

285 

30 

v  

63.0 

27 

13  8 

85 

6  9 

(d) 


e  

§2° 

212° 

390° 

K<7( 

1° 

7 

50° 

1100° 

c  

f 

5.09 

2.69 

2.90 

9  C 

IS 

s 

09 

3  28 

X  .  . 

1  ^ 

2  5 

3  5 

4  5 

[ 

5 

6  L 

7  5 

8  5 

V.  . 

3(1 

5 

3  92 

4  65 

5  30 

5  82 

6  ' 

10 

6  85 

7  25 

Ans.  y  =  2.5x1/2. 


XI,  §  177] 


EMPIRICAL  EQUATIONS 


241 


7.  Draw  each  of  the  following  curves: 

(a)  y  =  x1/2.  (6)    y  =  2x2. 

(c)  y  =  2s1/2.  (d)   y  =  3X3/2. 

(e)  y  =  8x~3/2.  (/)  y  =  1.5x2'3. 

(g)  y  =  9.2X-2/3.  (h)  y  =  log  x2/3. 

(i)    2/  =  10.  tf)    y  =  2-10*2. 

(fc)  y  =  10*/2.  (Q    y  =  10*+2. 

8.  Find  an  empirical  equation  connecting  the  x  and  y  values  given 
in  the  following  tables. 


(a) 


x  

0.2 

0.4 

0.6 

0.8 

•u  .  . 

3.18 

3.96 

5.00 

6.30 

Ans.  y  =  2.51(10"*). 


(d) 


x  

0.2 

0.4 

0.6 

0.8 

y  . 

5.8 

4.4 

3.4 

2.6 

X  

0 

14.4 

28.4 

42.2 

y  .  . 

180 

24 

3 

0.7 

X  

0 

41.4 

83.6 

126.2 

y  

180 

92 

46 

22 

9.  Given  age  in  years  and  diameter  in  inches  of  a  tree  If  feet  from 
the  ground  as  follows. 


Age  

19 

58 

114 

140 

181 

229 

Diameter  

3 

7 

13.2 

17.9 

24.5 

33 

Plot  the  data  and  determine  a  relation  of  the  form  y  =  kxn. 
10.  Given  age  in  years  and  height  in  feet  of  a  tree  as  follows : 


Age  

13 

34.4 

50.5 

218 

247 

Height  

13.4 

27.5 

38.4 

72.5 

73 

Plot  the  data  and  determine  a  relation  of  the  form  y  =  kxn. 

11.  Following  are  vapor  pressures,  in  mm.  of  mercury,  of  methyl 
alcohol  at  various  temperatures: 
17 


242 


MATHEMATICS 


[XI,  §  177 


t  

6 

13 

21 

30 

40 

'  42 

64 

100 

160 

260 

Represent  these  by  an  empirical  formula. 

12.    The  safe  load  W  in  tons  of  2000  Ibs.  for  a  beam  4  inches  wide  when 
the  distance  between  the  supports  is  12  feet  is  given  by 

W  =  KD\ 
where  D  is  the  depth  in  inches.     Find  K  from  the  following  table : 


D.  . 

10 

12 

14 

16 

18 

W 

1.85 

2.67 

3.63 

4.74 

6.00 

13.    Plot  a  curve  from  the  following  data,  find  its  equation,  and  esti- 
mate the  price  of  36-inch  pipe. 


Diameter  of  Sewer 
Pipe  .  .  . 

8 

10 

19! 

14 

16 

18 

20 

22 

24 

Price  in  i  per  linear 
ft  

?6 

7,1 

30 

36 

50 

68 

93 

125 

150 

14.  Plot  a  curve  from  the  following  data,  find  its  equation,  and 
estimate  the  pressure  for  a  velocity  of  110  miles  per  hour.  The  pressure 
is  given  in  pounds  per  square  foot  of  cross  section  of  the  first  car  in  a 
train  of  ten,  and  the  velocity  in  miles  per  hour. 


V.  .  .  . 

p  — 

32 
.97 

37 
1.35 

43 
1.80 

48 
2.25 

55 
3.32 

64 
4.18 

68 
4.83 

83 
6.75 

88 
7.72 

91 

8.37 

95 
9.01 

CHAPTER  XII 
THE  PROGRESSIONS 

178.  Arithmetic  Progression.     A  sequence  of  numbers  in 
which  each  term  differs  from  the  preceding  one  by  the  same 
number  is  called  an  arithmetic  progression  (denoted  by  A.  P.). 
The  common  difference  is  that  number  which  must  be  added 
to  any  term  to  obtain  the  next  one. 

To  determine  whether  or  not  a  given  sequence  is  an  arith- 
metic progression  we  find  and  compare  the  successive  differences 
of  consecutive  terms.  Thus 

3,  10,  17,24,31,  ••• 
is  an  A.  P.  in  which  the  common  difference  is  7. 

5,8,  11,  15,  18,  ••• 
is  not  an  A.  P. 

179.  Notation.     The  following  symbols  are  commonly  used 
to  denote  five  important  numbers,  called  elements,  which  are 
considered  in  connection  with  arithmetic  progressions. 

a  or  ai  =  the  first  term 

n  =  the  number  of  terms 
I  or  an  =  the  last  or  rith  term 

d  =  the  common  difference 
5  or  sn  =  the  sum  of  the  first  n  terms 

180.  Formulas.     If  the  terms  of  an  arithmetic  progression 
are  written  down  and  numbered  as  follows, 

Terms :  a,  a  +  d,  a  +  2rf,  a  +  3d,-  •  •  • 

Number  of  term :  1,       2     ,        3      ,        4      ,  ••• 

243 


244  MATHEMATICS  [XII,  §  180 

we  observe  that  the  coefficient  of  d  in  each  term  is  one  less  than 
the  number  of  the  term.  Hence  for  the  last  or  nth  term  we  have 
(1)  I  =  a  +  (n-  i)d 

We  may  write  the  progression  in  which  I  is  the  last  term  as 
follows: 

a,     a  +  d,     a  +  2d,     •  •  • ,     I  —  Id,     I  —  d,     I. 

The  sum  of  an  arithmetic  progression  is  found  by  adding  the 
n  terms  together: 

s  =  a  +  (a  +  d)  +  (a  +  2d)  +  •  •  •  +  (I  -  2d)  +  (I  -  d)  +  I. 
Inverting  the  order  of  the  terms 

s  =  I  +  (I  -  d}  +  (I  -  2d)  +  •  •  •  +  (a  +  2d)  +  (a  +  d)  +  a. 
By  addition  of  corresponding  terms,  we  have 

2s  =  (a  +  1)  +  (a  +  Z)  +  (a  +  Z)  +  •  •  •  +  (a  +  1)  +  (a  +  I) 
=  n(a  +  0- 


EXAMPLE.  Find  the  sum  of  an  arithmetic  progression  of  six  terms 
whose  first  term  is  4  and  whose  common  difference  is  2. 

Since  n  =  6,  we  have  I  =  4  +  5-2  =  14.  Hence  s  =  |(4  -}-  14) 
=  54. 

Given  any  three  of  the  elements  a,  n,  I,  d,  s,  either  of  the  other 
two  can  be  found  by  substituting  in  (1)  or  (2)  and  solving.  If 
n  is  to  be  found,  the  given  elements  must  be  such  that  the 
formula  will  be  satisfied  by  a  positive  integral  value  of  n. 

EXAMPLE.  Given  d  =  5,  Z  =  f,  s=—  -1/;  find  a  and  n.  Sub- 
stituting in  (1)  and  (2),  we  have 

3-      i-lfci       i\  15-1 

W  2  ~  a  +  2  (n  ~  l)'         ~  2"  ~  2 


XII,  §181]  THE  PROGRESSIONS  245 

Eliminating  a, 

n*  -  7n  -  30  =  0. 
Solving  for  n, 

n  =  10    or     -  3. 

The  value  n  =  —  3  is  inadmissible.  Substituting  n  =  10  in  (3),  we 
obtain  a  =  —  3.  Hence  n  =  10,  a  =  —  3,  and  the  arithmetic  pro- 
gression is  -  3,  -  2\,  -  2,  -  1|,  -  1,  -  i,  0,  i,  1,  H. 

181.  Arithmetic  Means.  The  terms  of  an  arithmetic  progres- 
sion between  the  first  and  last  terms  are  called  arithmetic  means. 
Between  any  two  numbers  as  many  arithmetic  means  as  desired 
can  be  inserted.  To  do  this  we  can  use  equation  (1)  to  compute 
the  common  difference  d,  for  a  and  /  are  known  and  n  is  two  more 
than  the  number  of  terms  to  be  inserted.  Then  the  required 
means  are.  a  +  d,  a  +  2d,  etc. 

The  problem  of  inserting  one  arithmetic  mean  between  two 
numbers  is  the  same  as  the  problem  of  finding  the  average  of 
two  numbers.  If  m  is  the  average  of  o  and  b,  then 


and  a,  m,  b  form  an  arithmetic  progression.     For  this  reason 
m  is  called  the  arithmetic  mean  of  a  and  6. 

EXAMPLE.  Insert  4  arithmetic  means  between  7  and  20.  Here 
a  =  7,  I  =  20,  n  =  6.  Substituting  these  values  in  (1),  we  have 
20  =  7  +  5-d,  whence  d  =  2|.  Hence,  the  required  means  are  9$, 
12i,  14f,  17|. 

EXERCISES 

Determine  which  of  the  following  suites  of  numbers  form  arithmetic 
progressions. 

1.    1,  7,  9,  12,  ».  2.   x,  x\  3x,  — 

3.    5,  8,  11,  14,  •••  4.   a  -  26,  a,  a  +  26,  ••• 

5.   3,  7,  11,  15,  •••  6.   4,  2,  0,  -  2,  ••• 

7.   2,  4,  6,  9,  -  8.   5,  3,  1,  -  1,  ••• 


246  MATHEMATICS  [XII    §  181 

Find  I  and  s  for  the  following  progressions : 
9.    -  2,  -  6,  -  10,  •••  to  17  terms. 

10.  3,  10,  17,  •••  to  50  terms. 

11.  5,  7.5,  10,  •••  to  36  terms. 

12.  2,  |,  V°>  4,  •••  to  48  terms. 

13.  Solve  formula  (1)  for  a,  n,  and  d  in  turn. 

14.  Solve  formula  (2)  for  a,  n,  and  I  in  turn. 

15.  Given  n  =  20,  a  =  1,  d  =  7 ;   find  I  and  s. 

16.  Given  n  =  1000,  I  =  500,  d  =  £ ;   find  a  and  s. 

17.  Given  n  =  16,  a  =  2,  Z  =  3 ;   find  d  and  s. 

18.  Given  a  =  2,  I  =  3,  s  =  100 ;   find  n  and  d. 

19.  Given  n  =  9,  a  =  1,  s  =  37;   find  d  and  Z. 

20.  Given  a  =  4,  d  =  0.1,  Z  =  8;   find  n  and  s. 

21.  Given  n  =  10,  d  =  0.2,  s  =  78 ;  find  a  and  Z. 

22.  Given  n  =  12,  I  =  -  3,  s  =  140 ;    find  a  and  d. 

23.  Given  d  =  3,  I!  =  22,  s  =  87 ;   find  a  and  n. 

24.  Given  a  =  8,  d  =  8,  s  =  80 ;   find  I  and  n. 

25.  Insert  3  arithmetic  means  between  1  and  17. 

26.  Insert  4  arithmetic  means  between  2  and  18. 

27.  Insert  5  arithmetic  means  between  3  and  38. 

28.  Insert  6  arithmetic  means  between  4  and  6. 

29.  Eight  stakes  are  to  be  set  at  equal  distances  between  the  two  cor- 
ners of  a  60  ft.  lot.     How  far  apart  must  they  be?     Ans.   6  ft.  8  in. 

30.  I  desire  to  close  up  one  side  of  crib  12  feet  4  inches  high,  with  6 
inch  boards.     I  have  just  21  boards.     I  desire  to  leave  a  1  inch  crack 
at  top  and  bottom.     How  far  apart  must  I  place  the  boards  to  have 
them  equally  spaced?  Ans.    1  inch. 

31.  At  the  end  of  each  year  for  10  years  a  man  invests  $200  on  which 
he  collects  annual  interest  at  6%.     Find  the  total  interest  received. 

Ans.   $540. 

32.  The  population  of  a  certain  town  has  made  a  net  gain  of  the  same 
number  of  people  each  year  for  the  last  30  years.     In  1893  it  was  1523 ; 
in  1906  it  was  2212.     What  was  it  in  1890  ?  in  1902  ?  in  1916  ?    Predict 
the  population  for  1925. 

33.  What  will  it  cost  to  erect  the  steel  work  of  a  20  story  building  at 
$3000  for  the  first  story  and  $250  more  for  each  succeeding  story  than 
for  the  one  below?  Ans.   $107500. 


XII,  §  183]  THE  PROGRESSIONS  247 

34.  I  drop  a  rock  over  a  cliff  400  ft  high.     How  long  before  I  hear  it 
strike  bottom  if  it  falls  16  ft.  the  1st  second,  48  ft.  the  2d  second,  80  ft. 
the  3d  second,  etc.,  and  sound  travels  1090  ft.  per  second  in  air? 

Ans.   5f  sec.  nearly. 

35.  A  ball  rolling  down  an  incline  goes  2  ft.  the  first  second  and  6  ft., 
10  ft.,  14  ft.,  respectively  in  the  next  three  seconds,  starting  from  rest. 
How  far  will  it  roll  in  15  seconds?  Ans.   450  ft. 

36.  A  clock  strikes  the  hours  and  also  1,  2,  3,  8,  respectively,  at  the 
quarter  hours.     How  many  strokes  does  it  make  in  a  day  ?    Ans.   422. 

37.  A  farmer  is  building  a  fence  along  one  side  of  a  quarter  section. 
The  post  holes  are  dug  one  rod  apart  and  the  posts  are  piled  at  the  first. 
How  far  will  he  walk  to  distribute  them  one  at  a  time  and  return  to  set 
the  first  one?  Ans.   20|  miles. 

38.  Find  the  sum  of  all  multiples  of  7  less  than  1000.     Ans.   71071. 

39.  Find  two  numbers  whose  arithmetic  mean  is  11  and  the  arith- 
metic mean  of  their  squares  is  157. 

40.  Show  that  if  an  A.  P.  has  an  odd  number  of  terms  the  middle  term 
is  the  arithmetic  mean  of  the  first  and  last. 

41.  If  the  sum  of  any  number  of  terms  of  the  A.  P.  8,  16,  24,  •••  be 
increased  by  1,  the  result  is  a  perfect  square. 

182.  Geometric  Progression.     A  sequence  of  numbers  in 
which  each  term  may  be  found  by  multiplying  the  preceding 
term  by  the  same  number  is  called  a  geometric  progression 
(denoted  by  G.   P.).     The  constant  multiplier   is   called   the 
common  ratio.    Thus 

3,  15,  75,  375,  ••• 
is  a  G.  P.  in  which  the  common  ratio  is  5. 

The  elements  of  a  geometric  progression  are  the  first  term  a 
or  oi,  the  number  of  terms  n,  the  last  or  nth  term  /  or  an,  and  the 
sum  s  or  sn  of  the  first  n  terms. 

183.  Formulas.     If  the  terms  of  a  geometric  progression  be 
written  down  and  numbered  as  follows, 

Term  :  a,  ar,  ar2,  ar3,  •  •  • 

Number  of  term:  1,    2,    3,    4  ,  ••• 


248  MATHEMATICS  XII,  §  183 

we  see  that  the  exponent  of  r  in  each  term  is  one  less  than  the 
number  of  the  term.     Hence  for  the  nth  or  last  term  we  have 

(4)  I  =  ar"-1 

The  sum  of  the  first  n  terms  of  the  preceding  geometric  pro- 
gression is 

s  =  a  +  ar  +  ar2  +  •••  +  arn~l 
Multiplying  both  sides  by  r, 

sr  =  ar  +  ar2  +  ar3  +  •  •  •  +  arn 

By  subtraction,  we  have 

sr  —  s  =  arn  —  a. 
Solving  the  last  equation  for  s,  we  get 


r  -  1  1  -  r 

From  (4)  we  obtain  rl  =  arn.     Hence  (5)  may  also  be  written 

a  -  rl 


(6) 


1  -  r 


The  two  fundamental  formulas  (4)  and  (6)  contain  the  five 
elements  a,  I,  n,  r,  s,  any  two  of  which  may  be  found  if  the 
other  three  are  given. 

EXAMPLE  1.     Find  s  if  a  =  1,  n  =  7,  r  =  4. 
Substituting  these  values  in  (5),  we  get 

47  -  1      16384  -  1 
S  =  T— ^     -3-      =5461. 

184.  Geometric  Means.  If  three  positive  numbers  are  in 
geometric  progression  the  middle  one  is  said  to  be  the  geo- 
metric mean  of  the  other  two.  It  is  easy  to  see  that  the  geo- 
metric mean  of  two  numbers  is  the  square  root  of  their  product. 
Thus  3  is  the  geometric  mean  of  2|  and  4. 

If  several  numbers  are  in  geometric  progression  all  the  inter- 


XII,  §  184]  THE  PROGRESSIONS  249 

mediate  terms  are  said  to  be  geometric  means  between  the  first 
and  last  terms.  We  can  insert  as  many  geometric  means  as  we 
wish  between  any  two  positive  numbers.  To  do  this  we  use 
equation  (4),  §  183,  to  compute  r;  a,  I,  and  n  being  known. 
Then  the  desired  means  are  ar,  ar2,  ar3,  etc. 

EXAMPLE.  Insert  three  geometric  means  between  4  and  16.  Since 
16  is  to  be  the  5th  term  we  have  a  =  4.  ar4  =  16,  whence  r4  =  4  and 
r  =  V2 ;  hence  the  five  terms  are  4,  4V2,  8,  8 V2,  16. 

EXERCISES 

Which  of  the  following  sets  of  numbers  form  geometric  progressions  ? 
1.   3,  -  6,  12,  -  24,  •••  2.   4,  6,  9,  13.5,  ••• 

3.   7,  18,  40,  •••  4.   8,  12,  18,  26,  ••• 

5.   a,  2a,  3a,  4a,  •••  6.   a,  a?,  a3,  ••• 

7.    V3  -  1,  V2.  V3  +  1,  -  8.   8,  4,  2,  1,  ••• 

9.   a,  -  a2,  a3,  -  a«,  •••  10.    \/2,  2,  2^2,  4,  — 

11.    \/2,  V6,  3^2,  •••  12.    9,  3,  1,  i  — 

13.  Solve  formula  (4)  for  a,  n,  and  r  in  turn. 

14.  Solve  formula  (6)  for  a,  I,  and  r  in  turn. 

15.  Given  a  =  2,  r  =  3,  n  =  12 ;   find  I  and  s. 

16.  Given  a  =  3,  r  =  5,  n  =  10 ;   find  I  and  s. 

17.  Given  a  =  4,  n  =  6,  s  =  252 ;   find  I  and  r. 

18.  I  =  486,  a  =  2,  n  =  6 ;  find  r  and  s. 

19.  Given  a  =  15,  r  =  3,  I  =  3645 ;  find  n  and  s. 

20.  Given  n  =  5,  r  =  \,  I  =  512 ;  find  a  and  s. 

21.  Insert  two  geometric  means  between  2  and  128. 

22.  Insert  3  geometric  means  between  2  and  162. 

23.  Insert  2  geometric  means  between  "N/2  and  108. 

24.  What  is  the  geometric  mean  between  a/6  and  6/a? 

25.  Find  the  6th  term  and  the  sum  of  the  series  2,  4,  8,  •••. 

26.  It  takes  32  nails  to  shoe  a  horse.     A  blacksmith  agrees  to  drive 
them  as  follows  :  2  cents  for  the  first,  4  cents  for  the  second,  8  cents  for 
the  third,  etc.     What  is  the  total  cost?  Ans.   $85,899,345.90 

27.  Find  the  amount  of  $500  in  5  years  at  6%  compounded  annually  j 
compounded  semiannually.  Ans.   $669.10;  $672.45 


250  MATHEMATICS  [XII,  §  184 

28.  In  how  many  years  will  $100  amount  to  $200,  interest  at  8% 
compounded  annually  ?     In  how  many  years  with  interest  at  6  %  com- 
pounded annually? 

Ans.    9  years  approximately ;  12  years  approximately. 

29.  A  man  promises  to  pay  $10,000  at  the  end  of  5  yr.     What  amount 
must  be  invested  each  year  at  6  %  compound  interest  so  that  at  the  end 
of  the  time  the  debt  can  be  paid? 

30.  A  premium  of  $104  is  paid  to  an  insurance  company  each  year 
for  10  years. 

What  is  the  value  of  these  amounts  at  the  end  of  the  time  if  accumu- 
lated at  3%  compound  interest? 

31.  A  premium  of  $91  is  paid  to  an  insurance  company  each  year  for 
10  years. 

What  is  the  value  of  these  amounts  at  the  end  of  the  time  if  accu- 
mulated at  3%  compound  interest? 

What  is  the  value  if  accumulated  at  4%  compound  interest? 

32.  An  insurance  company  agrees  to  pay  me  $1000  a  year  for  10 
years,  or  an  equivalent  cash  sum  to  myself  or  heirs  at  the  end  of  the 
period. 

Compute  the  equivalent  cash  sum  if  money  is  worth  6%  compound 
interest. 

33.  A  father  invests  $100  each  year  for  a  newborn  son,  beginning 
when  he  is  one  year  old. 

If  money  is  worth  4%  compounded  annually,  what  sum  is  due  the 
son  on  his  twenty-first  birthday  ? 

What  does  he  receive  on  his  twenty-first  birthday  if  the  amounts  in- 
vested bear  5%  compound  interest? 

34.  A  potato  cuts  into  4  parts  for  planting,   each  piece  produces 
5  good  sized  potatoes,  80  of  which  make  a  bushel.     If  I  plant  each 
year  all  that  I  raised  the  preceding  year,  how  many  bushels  of  potatoes 
will  I  have  at  the  end  of  the  fifth  year?     How  much  are  they  worth 
at  $4.00  per  bu.  ?  Ans.  $160,000. 

35.  One  kernel  of  corn  planted  produces  a  stalk  with  2  ears  with 
16  rows  each,  50  kernels  to  the  row.     Suppose  100  ears  make  a  bushel 
and  that  I  plant  each  year  one-half  of  all  that  I  raised  the  preceding 
year  and  that  one-half  of  the  kernels  grew  and  produced.     How  many 


XII,  §184] 


THE  PROGRESSIONS 


251 


bushels  would  I  have  at  the  end  of  the  fifth  year?     (Assume  two  kernels 
planted  the  first  year.) 

36.  I  have  one  sow.     Let  us  suppose  that  the  average  litter  of  pigs 
is  6,  sexes  equally  distributed,  and  that  I  keep  all  of  the  sows  each 
year  but  sell  all  the  others.     How  many  sows  in  the  sixth  generation? 
How  many  pigs  will  have  been  sold  after  I  have  disposed  of  1  /2  of  the 
last  or  5th  litter?  Am.  243;  363. 

37.  The  common  housefly  matures  and  incubates  a  new  litter  every 
3  weeks.     There  are  approximately  200  to  a  litter  evenly  distributed 
as  to  sex.     What  will  be  the  number  of  descendents  of  one  female  fly 
in  12  weeks?  Ans.  2  X  108. 

38.  Grasshoppers  hatch  yearly  a  brood  of  100  evenly  distributed 
as  to  sex.     Assuming  that  none  are  destroyed,  what  will  be  the  number 
of  descendants  of  one  female  grasshopper  at  the  end  of  5  years?  6  years? 

39.  The  apple  aphis  matures  and  incubates  in  10  days.     The  progeny, 
all  females,  are  5  in  number.     The  female  propagates  5  each  day  for 
30  days.     What  will  be  the  number  of  descendants  of  one  female  at 
the  end  of  30  days? 

40.  If  the  population  doubled  every  40  years,  how  many  descend- 
ants would  one  person  have  after  800  years?  Ans.  1,048,576. 

41.  Find  the  amount  of  money  that  could  profitably  be  expended 
for  an  overcoat  which  lasts  5  years  provided  it  saved  an  annual  doctor 
bill  of  $5,  money  being  worth  6%  compound  interest. 

42.  The  effective  heritage  contributed  by  each  generation  and  by 
each  separate  ancestor  according  to  the  law  of  ancestral  heredity  as 
stated  by  Galton  is  shown  in  the  following  table  from  Davenport. 


Generation  Back- 
ward. 

Eflective  Contribu- 
tion of  Each  Gen- 
eration. 

Number  of  Ancestors 
Involved. 

Effective  Contribution 
of  Each  Ancestor. 

1        

1/2 

2 

1/4 

2  

1/4 

4 

1/16 

3  

1/8 

8 

1/64 

4  

1/16 

16 

1/256 

5  

1/32 

32 

1/1024 

Compute  the  effective  contribution  of  the  last  20  generations.  The 
number  of  ancestors  involved  in  the  20th  generation  backward  and 
the  total  number  of  ancestors  involved.  The  effective  contribution  of 
each  ancestor  in  the  20th  generation  backward. 


252  MATHEMATICS  [XII,  §  185 

185.  Infinite  Geometric  Series.  A  geometric  progression  can 
be  extended  to  as  many  terms  as  we  please,  since  on  multiplying 
any  term  by  the  common  ratio  we  obtain  the  next  one.  Any 
series  which  has  no  last  term  and  can  be  indefinitely  extended  is 
called  an  infinite  series. 

Suppose  the  terms  of  a  geometric  series  are  all  positive.  If 
we  begin  at  the  first  and  add  term  after  term  the  sum  always 
increases.  If  r  >  1,  this  sum  becomes  infinite,  i.e.,  if  we  choose 
a  positive  number  N  no  matter  how  large  it  is  possible  to  add 
terms  enough  that  the  sum  will  exceed  N.  If  however  r  <  1, 
the  case  is  quite  different.  The  sum  does  not  become  infinite ; 
it  converges  to  a  limit,  i.e.,  it  is  possible  to  find  a  number  L  such 
that  the  sum  will  exceed  any  number  whatever  less  than  L,  but 
it  will  never  reach  L.  For  example  the  sum  obtained  by  adding 
terms  of  the  geometric  series 

1  +i+i+^  +  ••• 

in  which  r  =  -|,  will  never  reach  1.5,  but  terms  enough  can  be 
added  to  make  the  sum  exceed  any  number  less  than  1.5.  If, 
e.g.,  we  wish  to  make  the  sum  greater  than  1.49,  five  terms  are 
sufficient. 

A  geometric  series  in  which  r  <  1  is  called  a  decreasing  geo- 
metric series.  The  limit  to  which  the  sum  of  the  first  n  terms  of 
a  decreasing  geometric  series  converges  is  a/(\  —  r),  i.e.,  the  first 
term  divided  by  one  minus  the  ratio. 

For  by  (5)  §  183, 

8    =  a(l  -  r")  =      a  a     ^  ^ 

1  -  r          1  -  r      I  -  r 

Now  as  we  add  more  and  more  terms,  the  n  in  this  formula  gets 
larger  and  larger,  a  and  r  remain  fixed.  Since  r  <  1,  it  follows 
that  r2  <  r,  r3  <  r2,  etc.,  and  rn  converges  to  zero  when  n  is 
taken  larger  and  larger.  Therefore  the  second  term  on  the 
right  converges  to  zero,  and  sn  converges  to  a/(l  —  r).  This 


XII,  §  185]  THE  PROGRESSIONS  253 

limit  is  sometimes  called  the  "  sum  "  (although  strictly  it  is  not 
a  sum)  of  the  infinite  decreasing  geometric  series 

a  +  ar  +  or2  +  •  •  •, 
and  we  write 

(7)  s=^—'  r<1- 


EXAMPLE.  The  repeating  decimal  .666  •••  can  be  written  thus 
.6  +  .06  +  .006  +  •••.  It  is  therefore  an  infinite  geometric  series 
whose  first  term  is  .6  and  whose  common  ratio  is  .1.  Hence 

.6      _2 

fin  3- 

EXERCISES 
Find  the  sum  of  the  following  infinite  series  : 

1.  1+0.5  +0.25  +  -.  6.    1+I  +  H  +  -- 

2.  1  -0.5  +  0.25  -0.125  +  ••-.      7.   3  +  f  +  T35  +  •••. 

3.  l  +  i  +  i  +  —  .  8.    100  +  1  +0.01  +•••. 

4.  1  -  i  +  I  -  iV  +  ••••  9.   3  +  0.3  +  0.03  +  ••-. 

5..  1  +  f  +  |  +  ••-.  10.  0.23  +  0.023  +  0.0023  +  -. 
Find  the  value  of  the  following  repeating  decimals  : 

11.  .1111  -.  17.  .00032525  •••. 

12.  .2222  •••.  18.  .1234512345  •••. 

13.  .252252-.  19.  20.2020—. 

14.  1.2424  •••.  20.  5.312312  ••-. 

15.  2.53131  •••.  21.  6.4141  -. 

16.  2.3452345  ••-.  22.  3.214214  —. 


CHAPTER  XIII 
ANNUITIES* 

186.  Definitions.     Suppose  you  take  out  a  life  insurance 
policy  on  which  you  agree  to  pay  a  premium  of  $100  at  the  end 
of  each  year  for  10  years.     Such  an  annual  payment  of  money 
for  a  stated  time  is  termed  an  annuity.     Instead  of  paying  $100 
a  year  you  may  prefer  to  pay  $24  at  the  end  of  every  three 
months  or  $206  at  the  end  of  every  two  years.     In  any  case  the 
stated  amount  paid  at  the  end  of  equal  intervals  of  time  is  called 
an  annuity. 

Suppose  the  stated  sums  are  not  paid  when  due  and  that  after 
the  lapse  of  say  5  years  you  desire  to  pay  off  your  indebtedness 
with  interest  compounded.  The  sum  due  is  called  the  amount 
of  the  annuity  for  the  five  years. 

Suppose  you  buy  a  house  and  agree  to  pay  $1000  at  the  end 
of  each  year  for  4  years.  This  is  an  annuity.  An  equivalent 
cash  price  at  the  time  of  sale  is  called  the  present  value  of  the 
annuity. 

187.  Notation.     The  letter  r  stands  for  the  rate  of  interest, 
e.g.  6 ;  the  letter  f  ( =  r/100)  stands  for  the  annual  interest  on  one 
dollar,  e.g.  .06. 

The  symbol  S^  stands  for  the  amount  of  an  annuity  of  one 
dollar  paid  at  the  end  of  each  year  for  n  years. 


are  indebted  for  many  ideas,  methods,  and  exercises. 

254 


XIII,  §  189]  ANNUITIES  255 

The  symbol  S^  stands  for  the  amount  of  an  annuity  of  one 
dollar  paid  at  the  end  of  each  pth  part  of  a  year  for  n  years. 

The  symbol  a^\  stands  for  the  present  value  of  one  dollar  paid 
at  the  end  of  each  year  for  n  years. 

The  symbol  a^  stands  for  the  present  value  of  an  annuity  of 
one  dollar  paid  at  the  end  of  each  pth  part  of  a  year  for  n  years. 

188.  Amount  of  an  Annuity.  It  is  sufficient  to  consider  an 
annuity  of  one  dollar  since  the  amount  for  any  other  sum  will 
be  proportional  to  this. 

The  first  payment  of  one  dollar  made  at  the  end  of  the  first 
year  will  bear  interest  for  n  —  1  years,  and  at  the  end  of  the 
period  the  amount  due  will  be  (1  +  t)™"1.  The  second  payment 
will  bear  interest  for  n  —  2  years  and  will  increase  to  (1  +  i)n~2. 
The  next  to  the  last  payment  will  bear  interest  for  one  year  and 
will  increase  to  1  +  i.  The  last  payment  will  be  one  dollar  and 
it  will  bear  no  interest.  The  total  amount  S^,  due  at  the  end  of 
n  years  is  therefore 

1  +  (1  +  i}  +  (1  +  t)2  +  ».  +  (1  +  t)"-2  +  (1  +  i}"-1. 

In  this  geometric  progression  the  first  term  is  1,  the  last  term 
is  (1  +  i)n~S  and  the  ratio  is  1  +  i.  Substituting  these  values 
in  the  formula  (5)  §  183  for  the  sum  of  a  geometric  progression, 
we  find 


189.  Partial  Payments.  Suppose  that  the  payments  instead 
of  being  made  at  the  end  of  each  year  are  made  at  the  end  of 
each  pth  part  of  a  year  for  n  years.  Consider  an  annuity  of  one 
dollar. 

The  payment  to  be  made  at  each  payment  period  is  l/p.  The 
first  payment  will  bear  interest  for  n  —  \/p  years.  The  second 
payment  will  bear  interest  for  n  —  2/p  years,  and  so  on.  The 
next  to  the  last  payment  will  bear  interest  for  l/p  years.  The 


256  MATHEMATICS  [XIII,  §  189 

last  payment  will  bear  no  interest.     The  total  amount  due  is 
then 

i  +  1  (i  +  o*  +  -  a  +  o*  +  -  •  +  -  a  +  *r*  . 

p  P        P  P 

i 

In  this  geometric  progression  the  common  ratio  is  (1  +  i}v  , 
and  by  (5),  §  183,  the  sum  of  the  terms  is 


As  shown  in  §  145  for  the  square  root,  the  pth  root  of  1  +  i 
is  nearly  equal  to  1  +  i/p.  In  fact  it  is  customary  in  comput- 
ing the  amount  of  one  dollar  at  interest  compounded  p  times  a 
year,  to  use  1  +  i/p  instead  of  Vl  +  i-  See  §  217.  If  this  ap- 
proximate value  be  used  in  formula  (2),  the  right  member 
reduces  to 


^ 
which  shows  that  S^  is  approximately  equal  to  S^. 

EXERCISES 

1.  Find  the  amount  of  an  annuity  of  $200  for  10  years  at  3%  ;  4%  ; 
5%  ;  6%  ;  8%.  Ans.   For  3%  $2292.78 

2.  The  semiannual  premium  on  an  insurance  policy  is  $50.      Find 
the  amount  of  this  annuity  for  10  years  at  4%.  Ans.   $606.37 

3.  The  quarterly  premium  on  a  policy  is  $62.10.     Find  the  amount 
of  this  annuity  for  10  years  at  3%.  Ans.   $719.11 

4.  The  annual  rent  of  a  house  is  $480.     Find  the  amount  of  this 
annuity  for  20  years  at  6%.     Find  the  amount  if  the  rent  is  paid 
monthly.  Ans.   $17657.08 

5.  A  man  saves  and  at  the  end  of  each  year  for  40  years  deposits  $100 
in  a  savings  bank  which  pays  4%  compounded  annually.     Find  the 
amount.  Ans.   $9502.55 

6.  A  man  saves  $500  a  year  and  invests  savings  and  interest  in  bonds 
yielding  6%.     What  will  his  accumulations  amount  to  in  10,  15,  20, 
30  years?  Ans.   $6590.40 


XIII,  §  190]  ANNUITIES  257 

190.  Given  the  Amount  of  an  Annuity  to  find  the  Annuity. 
Let  the  annual  payment  be  x.  The  first  payment  made  one 
year  from  the  beginning  of  the  term  of  the  annuity  will  bear  in- 
terest for  n  —  1  years  and  will  increase  to  x(l  +  i)n~l.  Like- 
wise, the  second  will  increase  to  x(l  +  i}n~~,  the  third  to 
(1  +  z')"~3>  and  so  on,  while  the  last  payment  x  will  bear  no 
interest.  If  the  sum  of  the  amounts  due  at  the  end  of  n  years 
is  $1,  we  have 

x[(\  +  t)"-1  +  (1  +  i)"-2  +  •"  +  (1  +  i)  +  i]  =  1. 
The  expression  within  the  square  brackets  is  a  geometric  pro- 
gression of  n  terms  with  ratio  (1  +  i}  ;   hence,  by  (5),  §  183,  we 
have 


or 

'"(TT^"1' 

which  gives  the  annuity  whose  amount  after  n  years  is  $1.     This 
formula  for  x  may  be  written  symbolically  in  the  form 

(4)  x=-f. 

S*i 

EXERCISES 

1.  In  10  years  a  man  desires  to  be  worth  $30,000.     What  sum  must 
he  set  aside  yearly  to  realize  that  amount  if  money  is  worth  8%  ? 

2.  An  auto  truck  costing  $2000  lasts  5  years.     What  sum  must  be 
set  aside  annually  at  6%  to  replace  the  truck  when  worn  out? 

3.  An  automobile  costs  $1500  and  lasts  5  years.     What  is  the  equiva- 
lent annual  expenditure,  money  worth  6%? 

4.  A  city  decides  to  pave  some  of  its  streets.     For  this  purpose  bonds, 
bearing  6%  interest,  to  the  amount  of  $50,000  are  issued.     The  bonds 
are  due  in  10  years.     What  sum  must  be  collected  yearly  in  taxes  and 
invested  at  6%  to  pay  off  the  bonds  when  due? 


258  MATHEMATICS  [XIII,  §  191 

191.  Present  Value  of  an  Annuity.    The  present  value  of 
one  dollar  due  in  one  year  is     (1  +  *)-1, 
one  dollar  due  in  two  years  is  (1  +  i)~2> 

one  dollar  due  in  n  years  is        (1  +  i}~n. 

The  present  value  of  one  dollar  paid  at  the  end  of  each  year 
for  n  years  will  then  be 

(1  +  i)~l  +  (1+  i)-2  +  -  +  (1  +  ^ 

The  sum  of  this    geometric   progression  is  the  present  value 
sought.     Hence  the  present  value,  a$j-,,  of  an  annuity  of  $1  is 

(5)  a  .  -  (1  +  ^  -  (1  +  fl"1 

(1  +  i)-1  -  1 

Multiplying  numerator  and  denominator  by  1  +  i  we  find 


EXERCISES 

1.  A  man  buys  a  farm,  agreeing  to  pay  $1500  cash  and  $1500  at  the 
end  of  each  year  for  three  years.     What  would  be  the  equivalent  cash 
value  of  the  farm  if  money  is  worth  6%? 

2.  A  man  buys  a  farm,  agreeing  to  pay  $2000  cash  and  $2000  at  the 
end  of  each  year  for  ten  years.     What  would  be  the  equivalent  cash  value 
of  the  farm  if  money  is  worth  6%? 

3.  A  contractor  performs  a  piece  of  work  for  a  city  and  takes  bonds 
in  payment.     The  bonds  do  not  bear  interest,  and  are  payable  in  10 
equal  annual  installments  of  $2000,  the  first  payment  to  be  made  one 
year  from  date.     Money  being  worth  6%,  payable  annually,  what  is 
the  cash  value  of  the  bonds  on  the  date  of  issue  ? 

4.  Prove  that  the  present  value  of  one  dollar  paid  at  the  end  of  each 
pth  part  of  a  year  for  n  years  is 

1 


1  + 


(1  +  i)" 


and  show  that  this  is  approximately  equal  to  a^.     See  §  189. 


XIII,  §  192]  ANNUITIES  259 

5.  A  man  contracts  to  buy  a  house  paying  $200  every  three  months 
for  8  years.     Find  the  equivalent  cash  price,  money  being  worth  6%. 

6.  Find  the  cash  value  of  semiannual  payments  of  $500  for  5  years, 
money  being  worth  6%. 

192.  Cost  of  an  Annuity.  A  man  desires  to  provide  for  his 
family,  in  event  of  his  death,  an  annuity  of  $5000  a  year  for  20 
years.  What  amount  must  he  set  aside  in  his  will  to  provide 
for  this  annuity,  assuming  that  money  is  worth  6%. 

The  cost  of  an  annuity  of  one  dollar  per  year  for  n  years  is 
^/fi»  §§  187,  191.  Whence  the  cost  C,  of  an  annuity  of  P  dollars 
per  year  for  n  years  is 

(7) 


i 

From  this  we  compute  that  the  man  should  set  aside  in  his  will 
about  $57350. 

EXERCISES 

1.  What  will  be  the  cost  of  an  annuity  of  $500  a  year  for  10  years, 
money  being  worth  4%?  Ans.   $4055 

2.  A  man  agrees  to  pay  $700  a  year  for  5  years  for  a  house.     What  is 
the  cash  value  of  the  house,  money  being  worth  6%.      Ans.    $2948.66 

3.  A  man  agrees  to  pay  $700  a  year  for  20  years  for  a  farm.     What 
is  the  cash  value  of  the  farm,  money  being  worth  5%?     Ans.    $8723.55 

4.  A  man  70  years  old  has  $3000.     His  expectation  of  life  being  8 
years,  what  annuity  can  an  insurance  company  offer  him,  money  being 
worth  4%  ? 

5.  A  man  with  $10,000  pays  it  into  a  life  insurance  company  which 
agrees  to  pay  him  or  his  heirs  a  stated  sum  each  year  for  20  years. 
What  is  the  yearly  payment,  money  being  worth  4%? 

6.  A  man  buys  a  house  for  $4000.     What  annual  payment  will  can- 
cel the  debt  in  5  years,  money  being  worth  6%?  Ans.   $949.60 

7.  How  long  will  it  take  a  man  to  accumulate  $100,000,  by  saving 
$1000  a  year  and  investing  it  at  6%.  Ans.   33  yrs. 

8.  A  man  inherits  $20,000  which  is  invested  at  4%.     If  $1000  a 
year  is  spent,  how  long  will  the  inheritance  last.  Ans.   41  yrs. 


260  MATHEMATICS  [XIII,  §  193 

193.  Perpetuities.  In  the  previous  problems  treated  in  this 
chapter  the  payments  continued  over  a  fixed  number  of  years 
and  then  stopped.  The  annual  amount  expended  for  repairs 
on  a  gravel  road  does  not  stop  at  the  end  of  a  given  period,  but 
continues  forever.  Such  payments  constitute  an  endless  an- 
nuity, which  is  called  a  perpetuity.  Other  examples  are  the 
annual  repairs  on  a  house,  taxes,  annual  wage  for  a  flag  man, 
annual  pay  of  a  section  gang.  The  amount  of  an  annuity  would 
evidently  increase  indefinitely  as  time  went  on.  The  present 
value  of  a  perpetuity,  however,  has  a  definite  meaning.  The 
present  value  of  a  perpetuity  is  a  sum  which  put  at  interest  at 
the  given  rate  will  produce  the  specified  annual  income  forever. 
Denote  by  V  the  present  value  of  the  perpetuity  and  by  P  the 
annual  payment.  Then 

(8)  F  •  i  =  P. 

If  the  payments  are  made  every  n  years  instead  of  yearly, 
the  present  value  of  the  perpetuity  is  denoted  by  Vn ;  its  value 
will  be 

(9)  Vn  =  P[(l  +  i)-  +  (1  +  i)-2n  +  -  +  (1  +  i~)~pn  +  •••]. 

This  is  an  infinite  geometric  progression  whose  first  term  is 
P(l  +  i)-"  and  whose  ratio  is  (1  +  t)"".  Hence,  by  (7),  §  185, 
the  present  value  of  the  perpetuity  is 

(1  +  i)-»  P 


(10)  Vn  =  P 


1  -  (1  +  i)-»      (1  +  i)w  -  1 


EXERCISES 


1.  What  is  the  present  cash  value  of  a  perpetual  income  of  $1200  per 
year,  money  being  worth  6%  ?  Ans.   $20,000. 

2.  How  much  money  must  be  invested  at  6%  to  provide  for  an  in- 
definite number  of  yearly  renewals  of  an  article  costing  $24? 

3.  How  much  money  must  be  invested  at  4%  to  provide  for  the  pur- 
chase every  4  years  of  a  $1000  truck? 


XIII,  §  193]  ANNUITIES  261 

4.  What  is  the  cash  value  of  a  farm  that  yields  an  average  annual 
profit  of  $2400,  money  being  worth  6%? 

5.  The  life  of  a  certain  farming  implement  costing  $100  is  6  yrs. 
Find  what  sum  must  be  set  aside  to  provide  for  an  indefinite  number  of 
renewals,  money  being  worth  4%. 

6.  The  life  of  a  University  building  costing  $100,000  is  100  years. 
A  man  desires  to  will  the  University  enough  money  to  erect  the  building 
and  to  provide  for  an  indefinite  number  of  renewals.     How  much  must 
he  leave  the  institution? 


CHAPTER  XIV 
AVERAGES  * 

194.  Meaning  of  an  Average.     In  referring  to  a  group  of 
individuals,  a  detailed  statement  of  the  height  of  each  would 
take  considerable  time,  when  large  numbers  are  involved.     In 
comparing  two  or  more  groups,  such  a  mass  of  detail  might  fail 
to  leave  a  definite  impression  as  to  their  relative  heights.     What 
is  needed  is  a  single  number,  between  that  of  the  shortest  and 
that  of  the  tallest,  which  is  representative  of  the  group  with 
respect    to    the    character   measured.     Such   an    intermediate 
number  is  called  an  average. 

The  idea  of  an  average  is  in  use  in  everyday  affairs.  We 
hear  mentioned  frequently  such  expressions  as  the  average  rain- 
fall, the  average  weight  of  a  bunch  of  hogs,  the  average  yield 
of  wheat  per  acre  for  a  county  or  state,  the  average  wage,  the 
average  length  of  ears  of  corn,  the  average  increase  in  popula- 
tion, etc.  Often  these  expressions  are  used  with  only  an  indefi- 
nite idea  as  to  what  is  really  meant. 

In  this  Chapter  we  shall  discuss  some  of  the  averages  in  com- 
mon use,  and  we  shall  explain  the  circumstances  under  which 
each  is  to  be  used. 

195.  Arithmetic  Average.     The  arithmetic  average  is  the 

*  The  authors  of  this  book  are  indebted  for  many  ideas  in  this  Chapter  and  for  some 
of  its  methods  to  an  Appendix  by  H.  L.  RIETZ  to  E.  DAVENPORT,  Principles  of  Breeding, 
Ginn  and  Co.  Some  use  has  been  made  also  of  ZIZEK,  Statistical  Averages,  Henry  Holt 
and  Co. ;  PEARSON,  Grammar  of  Science;  BOWLEY,  Elements  of  Statistics;  and  SECRIST, 
Introduction  to  Statistical  Methods,  Macmillan/ 

262 


XIV,  §  196]  AVERAGES  263 

number  obtained  by  dividing  the  sum  of  the  measurements  taken 
by  the  number  of  those  measurements : 

/1N  .,,       ,.  sum  of  all  measurements 

arithmetic  average  =  —  — . 

number  of  measurements 

Thus,  if  we  measure  seven  ears  of  corn  and  find  their  lengths 
to  be  6,  7,  8,  9,  10,  11,  12  inches,  the  arithmetic  average  of  their 
lengths  is  9  inches.  Again,  the  arithmetic  average  of  6,  7,  8,  12, 
12  is  9.  This  example  shows  that  the  arithmetic  average  gives 
no  indication  of  the  distribution  of  the  items  and  that  there 
may  be  no  item  whose  measurement  coincides  with  the  average. 
However,  it  is  influenced  by  each  of  the  items,  and  it  is  easily 
understood  and  computed.  It  should  seldom  be  used  except  in 
conjunction  with  other  forms  of  averages.  When  used  alone  it 
should  be  for  descriptive  purposes  only. 

196.  Weighted  Arithmetic  Average.  In  measuring  the  given 
items  it  frequently  happens  that  there  are 

n\  items  with  the  same  measurement  /i, 
HZ  items  with  the  same  measurement  1%, 

nt  items  with  the  same  measurement  l^. 
Then  the  weighted  arithmetic  average  is  given  by  the  formula 

(2)    weighted  arithmetic  average  =  ni/1  +  n'2/2  +  -+"***. 

ni  +  w2  +  "•  +  nk 

In  the  simple  case  mentioned  above,  the  weighted  arithmetic 
average  gives  the  same  result  as  the  arithmetic  average.  Its 
chief  advantage  is  that  it  facilitates  computations.  For 
example  the  average  length  of  the  ears  of  corn  whose  individual 
lengths  are  6,  7,  8,  12,  12  can  be  found  as  follows  : 

average  length  =  1X6  +  1X7  +  1X8  +  2X12  = 

1+1 +1+2 
There  may  be  other  reasons,  however,  for  counting  one  item 


264  MATHEMATICS  [XIV,  §  196 

several  times.  Thus,  in  measurements,  an  item  that  is  known 
to  be  particularly  trustworthy  may  be  counted  doubly  or  triply. 
In  such  cases,  the  weighted  average  differs  from  the  arithmetic 
average. 

197.  The  Median.     If  we  arrange  the  numbers  representing 
the  measurements  of  the  items  in  order  of  magnitude,   the 
middle  number  is  called  the  median.     Thus,  the  median  length 
of  the  ears  of  corn  whose  lengths  are  6,  7,  8,  12,  12  inches  is  8 
inches.     In  case  there  are  an  even  number  of  items  the  median 
is  midway  between  the  two  middle  terms.     Thus  if  the  lengths 
of  four  ears  of  corn  are  6,  7,  9,  10  inches,  the  median  length  is 
8  inches.     There  is  no  ear  of  this  length  among  those  measured. 

The  median  is  often  used  because  it  is  so  easily  found.  Like 
the  arithmetic  mean,  it  gives  no  indication  of  the  distribution. 
It  can  be  used  even  when  a  numerical  measure  is  not  attached 
to  the  various  items.  For  example,  ears  of  corn  can  be  ar- 
ranged in  order  of  length  without  knowing  the  numerical  length 
of  any  ear ;  clerks  can  be  ranked  in  order  of  excellence ;  shades 
of  gray  may  be  arranged  with  respect  to  darkness  of  color ;  etc. 
The  median  is  the  central  one  of  a  group  and  is  unaffected  by  the 
relative  order  of  the  other  members  of  the  group.  Thus  it  is 
used  when  the  primary  interest  is  in  the  central  members. 

198.  The  Mode.     In  measuring  the  items  of  a  given  set  it 
may  happen  that  some  one  measurement  occurs  more  frequently 
than  any  other.     This  measurement  is  called  the  mode.     Thus, 
the  modal  length  of  six  ears  of  corn  whose  lengths  are  6,  7,  8,  12, 
12,  13  inches  is  12  inches.     A  set  of  measurements  may  have 
more  than  one  mode.     Thus  in  a  given  factory  there  might  be 
few  men  who  received  $2  per  day,  a  large  number  who  received 
$3,  a  small  number  who  received  $4,  and  a  large  number  who 
received  $5,  while  few  received  more  than  $5.    There  would  then 
be  two  modes  for  wages,  namely  $3,  and  $5. 


XIV,  §  199]  AVERAGES  265 

If  a  curve  be  plotted  using  measurements  as  abscissas  and  the 
number  of  items  corresponding  to  each  frequency  as  ordinates, 
the  mode  corresponds  to  the  maximum  ordinate  or  ordinates. 
(See  §  225.) 

Unlike  the  arithmetic  average,  and  the  median,  the  mode  is 
always  the  value  of  one  individual  measurement.  Extreme 
measurements  have  no  effect  upon  it. 

In  measuring  heights  of  men  we  might  place  all  those  over 
4.5  and  under  5.5  feet  at  5  feet.  For  this  distribution  the  mode 
would  necessarily  fall  at  one  of  the  integers.  If  we  arrange 
the  heights  in  three-inch  intervals  the  mode  might  not  appear 
as  an  integer,  although  it  would  be  near  the  mode  first  obtained. 
Thus  it  is  seen  that  the  mode  depends  upon  the  grouping  of  the 
measurements. 

The  existence  of  a  mode  shows  the  existence  of  a  type.  It  is 
the  mode  that  we  have  in  mind  when  we  speak  of  the  average 
height  of  a  three-year-old  apple  tree,  the  average  price  of  land,  or 
the  average  interest  rate. 

199.  The  Geometric  Average.  The  geometric  mean  of  two 
positive  numbers  has  been  defined  in  §  184.  By  analogy  we 
may  define  the  geometric  average  of  n  positive  numbers  as  the 
nth  root  of  their  product. 

If  a  growing  tree  doubles  its  diameter  in  20  years  what  is  its 
annual  percentage  rate  of  increase  ?  It  is  not  5%,  for  an  increase 
of  5%  a  year  would  give  the  following  diameters  at  the  end  of  the 
1st,  2d,  3d,  .  .  .,  20th  year 

which  would  give  a  final  diameter  greater  than  2.6d.     Evi- 
dently what  is  wanted  is  a  rate  r  such  that 

/  f    \20 

M    -i--L-   \       —   2 

I        J.  |  —  , 

v    100; 

whence  r  =  100(V2  —  1)  =  3.53+.     Hence  an  annual  increase 


266  MATHEMATICS  [XIV,  §  200 

of  about  3|%  will  double  anything  in  20  years.  The  geometric 
average  is  used  in  many  practical  affairs.  Knowing  the  average 
rate  of  growth  of  a  city  in  the  past  the  geometric  average  is  used 
to  predict  its  future  growth.  When  a  new  school  building  is 
being  designed,  for  example,  it  should  be  made  large  enough  to 
meet  the  future  growth  of  the  community  as  shown  by  this 
geometric  average. 

200.  Conclusion.  Given  a  set  of  items  numerically  measured 
or  not,  we  should  first  determine  whether  or  not  the  data  is  such 
as  to  warrant  any  kind  of  an  average.  Then  the  decision 
whether  one  or  another  kind  of  average  is  to  be  employed  de- 
pends upon  the  use  to  which  the  result  is  to  be  put.  If  the  data 
is  not  complete,  the  arithmetic  average  cannot  be  used.  If  we 
desire  to  characterize  a  type  in  such  a  case,  we  may  find  the 
mode,  for  which  the  data  need  not  be  complete. 

Frequently  it  is  best  to  make  use  of  more  than  one  kind  of 
average  in  describing  a  distribution.  It  must  be  remembered  that 
any  average  at  best  conveys  only  a  general  notion  and  never  con- 
tains as  much  information  as  the  detailed  items  which  it  repre- 
sents. 

EXERCISES 

1.  From  the  heights  of  the  members  of  your  class,  find  each  of  the 
following  kinds  of  average  height :  (a)  arithmetic,  (6)  median,  (c)  mode. 

2.  Determine  in  the  following  cases  which  average  is  meant :    mean 
daily  temperature ;  average  student ;  average  price  of  butter ;  average 
of  a  flock  with  respect  to  egg  production ;   average  salary  for  all  of  the 
teachers  of  a  state ;    average  number  of  bushels  of  corn  per  acre  for  a 
state  or  nation ;    normal  rainfall ;    average  number  of  pigs  per  litter ; 
average  number  of  hours  of  sunshine  per  day ;    average  speed  of  train 
between  two  stops ;  average  wind  velocity ;  mean  annual  rainfall ;  aver- 
age sized  apple ;    average  price  of  oranges  when  arranged  according  to 
sizes ;   average  date  of  the  last  killing  frost  in  the  spring ;  average  price 
of  land  per  acre  in  a  given  locality ;   average  gain  in  weight  per  day  of 
a  hog. 


XIV,  §  200]  AVERAGES  267 

3.  What  kind  of  an  average  is  meant  in  each  of  the  following  cases : 
one  fly  lays  on  an  average  120  eggs;    63%  of  the  food  of  bobolinks  is 
insects ;   every  sparrow  on  the  farm  eats  j  oz.  of  weed  seed  every  day ; 
the  average  gas  bill  is  $2  per  month ;  the  average  price  received  for  lots 
in  a  subdivision  was  $800;    repairs,  taxes,  and  insurance  on  a  house 
average  $100  per  year ;   the  average  amount  of  material  for  a  dress  pat- 
tern is  8  yards,  36  inches  wide ;  a  college  graduate  earns  on  an  average 
$1125  a  year,  while  the  average  yearly  earnings  of  a  day  laborer,  who 
has  no  more  than  completed  the  elementary  school,  is  $475. 

4.  Suppose  that  we  consider  5  millionaires  and  1000  persons  who  are 
in  poverty.     Find  the  arithmetic  average,  the  median,  and  the  mode  of 
the  wealth  of  this  group.     Which  best  portrays  conditions? 

5.  In  the  Christian  Herald  for  March  10,  1915,  p.  237,  it  is  stated  that : 
"The  average  salary  of  ministers  of  all  denominations  is  $663.     The 
few  large  salaries  bring  up  the  average."     Which  average  is  used  here? 
Is  it  the  best  to  portray  conditions?     Is  the  result  too  high  or  too  low 
to  represent  conditions  properly? 

6.  Compute  for  the  members  of  your  family  the  mean  age,  and  arith- 
metic average.     Is  there  a  mode? 

7.  On  a  given  street  ascertain  the  number  of  houses  per  block  for  5 
blocks.     Find  the  arithmetic  average  and  the  median.     Is  there  a  mode  ? 

8.  On  a  given  business  street  ascertain  the  number  of  stories  of  each 
business  house  for  one  block.     Find  the  arithmetic  average  and  the 
median.     Is  there  a  mode? 

9.  Proceed  as  in  Ex.  8  for  a  residence  street.     Is  there  a  mode  ? 

10.  In  4  years  the  number  of  motorists  killed  at  railroad  crossings 
doubled.     Find  the  annual  rate  of  increase,  using  the  geometric  average. 

Ans.   19%. 

11.  If  in  the  last  20  years  the  number  of  deaths  in  the  U.  S.  due  to 
consumption  has  increased  50%,  find  the  annual  rate  of  increase,  using 
the  geometric  average.  Ans.   2%. 

12.  Land  increased  in  value  from  $40  to  $150  per  acre  from  1890  to 
1915.     What  was  the  average  yearly  increase? 

13.  Find  the  average  (arithmetic)  word,  sentence,  and  paragraph 
length,  of  some  one  of  the  writings  of  Longfellow,  Holmes,  Whittier, 
Poe ;  of  some  short  story ;  of  some  newspaper  article. 

14.  The  total  of  the  future  years  which  will  be  lived  by  100,000 


268  MATHEMATICS  [XIV,  §  200 

persons  born  on  the  same  day  are  5,023,371.  If  the  total  number  of 
-years  to  be  lived  is  divided  by  the  number  of  persons  the  quotient  will 
be  the  average  number  of  future  years  to  be  lived  by  each  person. 
What  kind  of  an  average  is  this  ?  What  average  age  does  it  give  ? 

15.  Out  of  100,000  males  born  alive  on  the  same  date  about  one-half, 
namely  50,435,  attain  age  59.  This  is  then  an  average  age  attained. 
What  kind  of  an  average  is  it? 


CHAPTER  XV 
PERMUTATIONS  AND   COMBINATIONS 

201.  Introduction.  In  how  many  ways  can  I  make  a  selec- 
tion of  two  men  to  do  a  day's  work  if  there  are  3  men  available 
for  the  forenoon  and  4  for  the  afternoon?  Having  hired  one 
man  for  the  forenoon,  I  can  hire  any  one  of  4  for  the  afternoon, 
and  since  this  is  true  for  each  of  the  three,  there  are  3  X  4  =  12 
ways  of  making  the  selection.  This  reasoning  is  general ;  that 
is,  it  does  not  depend  upon  the  special  properties  of  the  numbers 
3  and  4.  Hence  we  see  that  if  there  are  p  ways  of  doing  a  first 
act,  and  if  corresponding  to  each  of  these  p  ways  there  are  q  ways 
of  doing  a  second  act,  then  there  are  pq  ways  of  doing  the  sequence 
of  two  acts  in  that  order. 

It  is  evident  also  that  this  principle  applies  to  a  sequence  of 
more  than  two  acts  and  we  may  say, 

If  there  are  p  ways  of  doing  a  first  act;  and  if  after  this  has 
been  done  in  any  one  of  these  p  ways  there  are  q  ways  of  doing  a 
second  act;  etc.;  and  if  after  all  but  the  last  of  the  sequence  have 
been  done  there  are  r  ways  of  doing  the  last  act,  then  all  the  acts  of 
the  sequence  can  be  done  in  the  given  order  in  pq  •••  r  ways. 

EXERCISES 

1.  With  4  acids  and  6  bases,  how  many  salts  can  a  student  make? 

2.  A  ranchman  has  5  teams,  4  drivers,  and  3  wagons.     In  how  many 
ways  can  he  make  up  one  outfit? 

3.  There  are  6  routes  from  Chicago  to  Seattle,  4  from  Seattle  to  Port- 
land, 3  from  Portland  to  San  Francisco.     How  many  ways  are  there  of 
going  from  Chicago  to  San  Francisco  via  Seattle  and  Portland? 

269 


270  MATHEMATICS  [XV,  §  202 

202.  Combinations  and  Permutations.    A  group  of  things 
selected  from  a  larger  group  is  called  a   combination.     The 
things  which  constitute  the  group  are  called  elements.     Two 
combinations  are  alike  if  each  contain  all  the  elements  of  the 
other  irrespective  of  the  order  in  which  they  appear.     Two 
combinations  are  different  if  either  contains  at  least  one  element 
not  in  the  other. 

A  permutation  of  the  elements  of  a  group  or  combination,  or 
simply  a  permutation,  is  any  arrangement  of  these  elements. 
Two  permutations  are  alike  if,  and  only  if,  they  have  the  same 
elements  in  the  same  order.  Thus,  eat,  tea,  and  ate  are  the  same 
combination  of  three  letters  a,  e,  t ;  but  they  are  different  per- 
mutations of  these  three  letters. 

203.  Number  of  Permutations.     The  number  of  permuta- 
tions of  three  elements  taken  all  at  a  time  is  6,  as  may  be  seen  by 
writing  them  down  and  counting  them : 

abc,  acb,  bac,  bca,  cab,  cba. 

The  number  of  permutations  of  4  elements  taken  2  at  a  time  is 
12.     Thus, 

ab,  ac,  ad ;  ba,  be,  bd ; 

ca,  cb,  cd ;  da,  db,  dc. 

If  the  number  of  elements  is  large  the  process  of  counting  is 
tedious.  It  is  possible  to  derive  general  formulas  for  the  num- 
ber of  permutations  of  any  number  of  elements  by  which  the 
number  can  be  easily  computed. 

204.  Permutations  of  n  Things.     A  rule  for  the  number  of 
permutations  of  n  things  taken  all  at  a  time  is  easily  deduced 
by  means  of  the  principle  of  §  201 .     We  have  n  elements  and  n 
places  to  fill.     We  may  think  of  a  row  of  cells  numbered  from 
1  to  n. 


XV,  §  205]       PERMUTATIONS  AND  COMBINATIONS       271 


1 

2 

3 

4 

n 

The  first  cell  can  be  filled  in  n  different  ways  and  after  it  has 
been  filled  the  second  cell  can  be  filled  in  n  —  1  ways.  There- 
fore the  first  two  can  be  filled  in  n(n  —  1)  ways.  When  they 
have  been  filled  in  any  one  of  these  possible  ways  the  third  cell 
can  be  filled  in  (n  —  2)  ways.  Therefore  the  first  three  cells 
2an  be  filled  in  n(n  —  l}(n  —  2)  ways.  Continuing  thus  we 
see  that  the  first  k  cells  (k  <  n)  can  be  filled  in  n(n  —  l}(n  —  2) 
•••  (n  —  k  +  1)  ways,  and  that  all  the  n  cells  can  be  filled 
in  n(n  —  l}(n  —  2)  -"2  •  1  ways.  This  product  of  all  the 
natural  numbers  from  1  to  n  is  called  factorial  n,  and  is  denoted 
by  n  !  or  \n.  Thus,  2  !  =  2,  3  !  =  6,  4 !  =  24,  10 !  =  3,628,800. 
Therefore, 

The  number  of  permutations  of  n  things  taken  all  at  a  time  is 
factorial  n. 

For  example,  4  horses  can  be  hitched  up  in  24  ways  ;  10  cows 
can  be  put  into  10  stanchions  in  3,628,800  ways. 

By  the  same  reasoning  the  number  of  permutations  of  n 
things  k  at  a  time  (k  ^  n)  is  the  number  of  ways  that  k  cells 
can  be  filled  from  n  things.  The  symbol  nPt  is  used  to  denote 
this  number.  Then,  as  shown  above, 

(1)  nPk  =  n(n  -  l)(n  -  2)  •••  (n  -  k  +  1) 

To  remember  this  formula,  note  that  the  first  factor  is  n  and  the 
number  of  factors  is  k.  Thus  B^3  =  5  •  4  •  3  =  60.  The 
number  of  ways  in  which  4  stanchions  can  be  filled  out  of  a  herd 
of  10  cows  is  10P4  =  10  •  9  •  8  •  7  =  5040.  In  this  notation 
we  should  write  for  the  number  of  permutations  of  n  things 
all  at  a  time 

(2)  nPn  =  n\ 

205.  Repeated  Elements.  The  above  reasoning  assumes  that 
the  elements  are  all  distinct.  If  some  of  the  n  elements  are  alike, 


272  MATHEMATICS  [XV,  §  205 

the  number  of  distinguishable  permutations  is  less  than  n  \ 
For  example,  the  number  of  distinct  permutations  that  can  be 
made  out  of  the  7  letters  of  the  word  reserve  is  not  7 !  The 
number  of  permutations  of  the  7  characters  ri,  ei,  s,  62,  r2,  v,  63 
is  indeed  7  ! ;  but  when  the  subscripts  are  dropped  the  permuta- 
tions TI  e\  s  e^rzv  63  and  r^  £2  s  e3  r\  v  e\  become  identical. 

Let  x  be  the  number  of  different  permutations  of  the  letters 
of  the  word  reserve.  For  each  of  these  x  there  will  be  2  !  per- 
mutations of  the  characters  r\  e  s  e  r<z  v  e  and  for  each  of  these 
x  •  2 !  there  will  be  3 !  permutations  of  the  characters  r\  c\- 
s  cz  rz  v  63,  making  x  •  2  !  3  !  in  all.  It  follows  that 

x  •  2  I  3  !  =  7  !  and  x  =  -^~ 
2!3! 

This  reasoning  can  be  extended  to  show  that  the  number  of 
distinguishable  permutations  of  n  elements  of  which  p  are  alike, 
q  others  are  alike,  etc.,  •••,  r  others  are  alike,  is  equal  to 


(3) 


n ! 


p !  q  I  •••  r  I 


EXERCISES 


1.  How  many  3-letter  words  can  be  formed  from  the  letters  a,  p,  <? 
How  many  2-letter  words  ?     How  many  of  each  are  used  in  the  English 
language  ? 

2.  How  many  different  2-digit  numbers  can  be  made  from  the  ten 
digits  0,  1,  2,  •••,  9 ?     How  many  if  repetitions  are  allowed ?     How  many 
of  these  are  used? 

3.  Find  the  number  of  permutations  of  the  letters  in  each  of  the  fol- 
lowing words :     (a)  degree,   (6)  natural,   (c)  Indiana,   (d)  Mississippi, 
(e)    Connecticut,    (/)    Kansas,    (g)    Pennsylvania,    (h)     Philadelphia, 
(i)  Onondaga,  (j)  Cincinnati. 

4.  In  how  many  ways  can  a  pack  of  52  cards  be  dealt  into  four  piles 
of  13  each? 

5.  With  15  players  available,  in  how  many  ways  can  the  coach  fill 
the  various  positions  on  a  baseball  team? 


XV,  §  206]       PERMUTATIONS  AND  COMBINATIONS       273 

6.  How  many  different  signals  of  two  flags,  each  one  above  the  other, 
can  be  made  with  five  different  colored  flags  ? 

7.  How  many  different  sounds  can  be  made  by  plucking  the  five 
strings  of  a  banjo  one  or  more  at  a  time? 

8.  How  many  football  signals  can  be  given  with  four  numbers,  no 
repetitions  being  allowed? 

9.  In  how  many  ways  can  four  fields  be  cropped  with  corn,  oats,  wheat, 
and  clover,  one  field  to  each? 

10.  A  seed  store  offers  12  varieties  of  garden  seeds.     My  garden 
has  8  rows.     In  how  many  ways  can  I  plant  one  row  of  each  variety 
selected? 

11.  In  how  many  ways  can  a  gardener  plant  2  rows  of  lettuce,  3  of 
onions,  3  of  beans,  4  of  potatoes,  if  his  garden  has  12  rows? 

12.  How  large  a  vocabulary  could  be  formed  with  9  letters,  no  repe- 
titions   being    allowed?     How    many    with    ten?     How    many    with 
twenty-six?     (There  are  about  100,000  words  in  Webster's  dictionary. 
The  average  man  has  a  vocabulary  of  less  than  5000  words.) 

206.  Combination  of  n  Things  k  at  a  Time.  The  symbol 
nCk  or  (2)  is  used  to  denote  the  number  of  different  combinations 
(§  202)  that  can  be  made  from  n  elements  taken  k  at  a  time. 

A  combination  of  k  elements  can  be  arranged  into  k  !  permuta- 
tions of  these  elements.  That  is,  there  are  k  I  times  as  many 
permutations  as  there  are  combinations  of  k  elements  taken  all 
at  a  time.  Whence 

nPk  =  k\nCk. 

Making  use  of  the  value  of  nPt,  (1),  §  203,  and  solving  for  nCk 
we  have, 

(M  r    _"("  ~l)(n  -2)--(n  -fc  +  1) 

1.2.3-* 

To  remember  this  formula  note  that  the  first  factor  of  the  nu- 
merator is  n,  and  that  there  are  k  factors  in  the  numerator  and 
k  in  the  denominator. 

Another  useful  form  of  this  result  is  obtained  by  multiplying 


274  MATHEMATICS  [XV,  §  206 

both  numerator  and  denominator  of  (4)  by  (n  —  k}(n  —  k  —  1) 
(n  -k  -  2)  •••  2  •  1.  This  gives 

(5)  nCk  =         '1-TW 

k  i(n  —  k)  I 

We  note  that  the  interchange  of  Jc  and  n  —  k  leaves  (5)  un- 
altered and  hence  conclude  that 

(6)  n^n-k    =   nCk- 

This  is  what  we  should  expect  when  we  think  that  the  numbers 
of  ways  that  k  things  can  be  selected  from  a  group  of  n  must  be 
the  same  as  the  number  of  ways  that  n  —  k  can  be  rejected. 

EXERCISES 

1.  From  a  pack  of  52  cards  how  many  different  hands  can  be  dealt? 

2.  How  many  combinations  of  5  can  be  drawn  from  42  dominoes? 

3.  How  many  different  tennis  teams  can  be  made  up  from  6  players 
(a)  singles ;  (6)  doubles  ? 

4.  How  many  straight  lines  can  be  drawn  through  8  points,  no  three 
of  which  lie  on  a  straight  line  ?     How  many  circles  ? 

5.  How  many  diagonals  has  a  convex  polygon  of  n  vertices? 

Ans.  nCi.  —  n. 

6.  Two  varieties  of  corn  are  planted  near  each  other.     How  many 
varieties  will  be  harvested?  Ans.   zCz  +  2. 

7.  If  four  varieties  of  oats  are  sown  near  each  other,  how  many  varie- 
ties will  be  harvested?  Ans.   4^2  +  4. 

8.  A  starts  with  two  kinds  of  pure-bred  chickens.     How  many  kinds 
will  he  have  at  the  end  of  the  third  hatching  if  all  stock  is  sold  when 
one  year  old?  Ans.   «Cz  +  6. 

9.  In  how  many  ways  can  15  gifts  be  made  to  3  persons,  5  to  each? 

Ans.   isCe  •  i0C5. 

10.  In  how  many  ways  can  15  gifts  be  made  to  3  persons,  4  to  A, 
5  to  B,  6  to  C?  Ans.   630,630. 

11.  Given  (a)  nC2  =  45;   (6)  BC2  =  190;   (c)  nC2  =  105;  find  n. 

12.  In  how  many  different  ways  can  500  ears  of  corn  be  selected  from 
505  ears? 

13.  Compute:   (a)  loooCW;   (&)  mCm;   (c)  10002^10000. 


CHAPTER  XVI 

THE  BINOMIAL  EXPANSION— LAWS   OF 
HEREDITY 

207.  Product  of  n  Binomial  Factors.  If  the  indicated  mul- 
tiplications are  performed  and  terms  containing  like  powers  of  x 
are  collected, 


(1)     (x  +  Oi)(a;  +  a2)(.r  +  a,)(x  +  o4)  •  •  •  (x  +  an) 

=  Xn  +  CiX—1   +  CZXn~*  +  C3Xn~3  +   •  •  •    +  Cn-iX  +  Cn 

in  which  the  coefficients  have  the  following  values: 

Ci  =  01  +  a2  +  «3  +  •  •  •  +  an. 
The  number  of  these  terms  is  n. 
C2  =  Oi02  +  •  •  •  aian  +  aza3  +'•••+  a3a4  +  •  •  •  +  an-i«n. 

The  number  of  these  terms  is  the  number  of  combinations 
that  can  be  made  from  n  a's,  2  at  a  time,  i.  e.,  nC2. 

Cs  =  aia2a3  +  aia2a4  +  •  •  •  +  a2a3a4  +  •  *  •  +  an_2an_iOn. 

The  number  of  these  terms  is  the  number  of  combinations 
that  can  be  made  from  n  a's,  3  at  a  time,  i.  e.,  nC3. 

d  =  Oia2a3a4  +  aia2a3a6  +  •  •  •  +  an_3on_2an_ian. 
The  number  of  these  terms  is  nCt. 

Cr  =  aiO2o3  •  •  •  ar  + 
The  number  of  these  terms  is  «Cr. 

Cn  =  aidzds  • '  •  an,  and  consists  of  one  term. 

275 


276  MATHEMATICS  [XVI,  §  208 

If  now  each  of  the  a's  be  replaced  by  y,  it  is  evident  that, 

Ci  =  ny,         Cz  =  nC2y*,         C3  =  nC3y3, 
r   -     r  if  r    -  nn 

\sr   —   n^ry  ,  >  ^n   —    y    j 

and  therefore 

(2)       (x  +  y)n  =  xn  +  nxn~ly  +  nC2x"-2z/2  +  nC3xn~3y3  +  •  •  • 

+  nCrxn~ryr  +  ••••+  nxyn~l  +  yn. 

This  is  known  as  the  binomial  expansion,  or  binomial  formula. 

208.  Binomial  Theorem.  If  x  and  y  are  any  real  (or  imagin- 
ary) numbers  and  if  n  is  a  positive  integer,  then  the  binomial 
formula  (2)  is  valid.  The  following  observations  will  be  of 
value. 

(1)  The  exponent  of  x  in  the  first  term  is  1  and  decreases  by 
1  in  each  succeeding  term. 

(2)  The  exponent  of  y  in  the  second  term  is  1  and  increases 
by  1  in  each  succeeding  term. 

(3)  The  coefficient  of  the  first  term  is  1,  that  of  the  second 
term  is  n.     The  coefficient  of  any  term  can  be  found  from  the 
next  preceding  term  by  multiplying  the  coefficient  by  the  exponent 
of  x  and  dividing  by  one  more  than  the  exponent  of  y. 

(4)  The  (r  +  l)th  term  is  nCrxn-ryr,  i.  e., 

n(n  -  l)(w  -  2)  • •  •  (n  -  r  +  1) 

__i 1J ' : '        '  ~n— r,,r 

r! 

The  coefficient  of  this  (r  +  l)th  term  is  the  product  of  the 
first  r  factors  of  factorial  n,  divided  by  factorial  r. 

(5)  The  sum  of  the  exponents  of  x  and  y  in  any  term  is  n. 

(6)  The  number  of  terms  is  n  -\-  1. 

To  prove  the  rule  in  statement  (3)  apply  it  to  the  (r  +  l)th 
term, 

n   .  ~n—r*,r 
n^r   X        y  . 


XVI,  §  210]          THE  BINOMIAL  EXPANSION 


277 


It  gives 


n  —  r      n(n  — 


r  +  1  r! 

n(n  -  l)(n  -  2)  •  •  •  (n  -  r) 


—  2)  •  •  •  (n  —  r  +  1)     n  —  r 
'  r  +1 


(r  +  l)l 

but  this  is  precisely  nCV+i,  which  was  to  be  proved. 

209.  Binomial  Coefficients.  The  coefficients  in  the  bi- 
nomial expansion  are  called  binomial  coefficients.  Their  values 
are  given  in  the  following  table  for  a  few  values  of  n.  This 
table  is  called  Pascal's  triangle. 

TABLE  OP  BINOMIAL  COEFFICIENTS,  nCr. — PASCAL'S  TRIANGLE 


r=0 

r  =  l 

r=2 

r=3 

r=4 

r=5 

r=6 

r=7 

r  -8 

r=9 

r=lO 

r=ll 

n=  1 

1 

1 

n=  2 

1 

2 

1 

n=  3 

1 

3 

3 

1 

n=  4 

1 

4 

6 

4 

1 

n=  5 

1 

5 

10 

10 

5 

1 

n=  6 

1 

6 

15 

20 

15 

6 

1 

n=  7 

1 

7 

21 

35 

35 

21 

7 

1 

n=  8 

1 

8 

28 

56 

70 

56 

28 

8 

1 

n=  9 

1 

9 

36 

84 

126 

126 

84 

36 

9 

1 

n  =  10 

1 

10 

45 

120 

210 

252 

210 

120 

45 

10 

1 

n  =  ll 

1 

11 

55 

165 

330 

462 

462 

330 

165 

55 

11 

1 

etc. 

etc. 

etc. 

NOTE.  If  any  number  in  the  table  be  added  to  the  one  on 
its  right,  the  sum  is  the  number  under  the  latter. 

210.  Sum  of  Binomial  Coefficients.  A  great  many  uses  for 
binomial  coefficients  and  a  great  many  relations  among  them 
have  been  discovered.  Two  of  these  are  as  follows. 

(1)  The  sum  of  the  binomial  coefficients  of  order  n  is  2n.  We 
verify  from  the  above  table  that 

1  +  1-  21;       1+2  +  1=  -22;       1  +  3  +  3  +  1  =  23;       etc. 
To  prove  it  for  any  value  of  n,  put  x  =  1  and  y  =  1,  in  the 


278  MATHEMATICS  [XVI,  §  211 

binomial  formula: 

(1    +   l)ra    =    1    +  nCl   +  nC2  +    •  •  •    +  »Cn-l   +  nCn 

which  proves  the  statement. 
Transposing  1,  we  have 

«Ci   +  nC2  +  nC3  +    •  '  •    +  nCn    =   2n   -I 

i.  e.,  the  total  number  of  combinations  of  n  things  taken  1,2,  3, 
"  ' ,  n,  at  a  time  is  2n  —  1. 

(2)  The  sum  of  the  odd  numbered  coefficients  is  equal  to  the 
sum  of  the  even  numbered  ones  and  each  is  2""1. 

We  verify  from  the  table,  that 

1  =  1,         1  +  1=2,         1+3  =  3  +  1, 
1+6  +  1=4  +  4,         etc. 

To  prove  it  for  any  value  of  n,  put  x  —  1,  y  =  —  1,  in  the  bi- 
nomial formula: 

(1    -    1)"    =    1    -  nd   +  nC2   -  nC3  +  nC4   -    •  •  •    ±  nCn 

whence 

1    +  nC'2   +  nC*  +    •  •  •     =   nC\   +  nCs  +  nCg  +    '  •  •  . 

211.  Use  of  the  Binomial  Theorem.  In  expanding  a  bi- 
nomial with  a  given  numerical  exponent,  the  student  is  urged 
to  find  thq  successive  coefficients  by  using  the  statement  (3)  §  208, 
and  not  by  substitution  in  a  formula.  This  is  illustrated  in  the 
following  examples. 

EXAMPLE  1.     Expand  (2z  —  3?/)5. 

(2x  -  Si/)5  =  (2x)»  +  5(2a;)«(-  3?/)1  +  10(2z)3(-  Sy)2 

+  10(2x)2(-  3?/)3  +  5(2z)(-  ZyY  +  (-  3y)fi. 


XVI,  §  211]          THE  BINOMIAL  EXPANSION  279 

The  coefficients  are  computed  mentally  as  follows, 

the  3d   coefficient  from  the  2d   term  :    5X4/2  =  10, 
the  4th          "  "       "    3d    term  :  10  X  3/3  =  10, 

the  5th          "  "       "    4th  term  :  10  X  2/4  =  5, 

the  6th         "  "       "    5th  term  :    5X1/5  =  1. 

Simplifying  the  terms,  we  have 

(2x  -  3?/)5  =  32z8  -  240x4t/  +  720z32/2  -  2160z2?/3  +  810zy4  -  243t/5. 
EXAMPLE  2.     Expand  (3  —  |)6. 

(3  -  |)6  =  3«  +  6(3)6(-  i)i  +  15(3)<(-  W  +  20(3)«(-  i)3 

+  15(3)2(-  |)<  +  6(3)'(-  I)5  +  (-  £)6. 
The  coefficients  are  computed  as  follows: 

6X5/2  =  15,         15  X  4/3  =  20,        20  X  3/4  =  15,        etc. 
Simplifying,  we  have 

+ 

729.  729. 

303.75  67.5 

8.4375  0.5625 


0-015625  797.0625 

1041.203125 
797.0625 
(2|)6  =    244.140625 

EXAMPLE  3.     Expand  (a  +  b  +  c)3. 
[(a  +  6)  +  c]3  =  (a  +  6)3  +  3  (a  +  6)2c  +  3  (a  +  6)c2  +  c3 

=  a3  +  3a26  +  3afe2  +  b3  +  3  (a2  +  2ab  +  62)c 

+  3(o  +  b)c2  +  c3 
=  a3  +  63  +  c3  +  3a26  +  3a2c  +  362c  +  362a  +  3c2a 

+  3c26  +  6o6c. 
EXERCISES 

Expand  the  following  expressions  by  the  binomial  theorem. 

1.    (x  +  3)5.  2.    (y  -  4)«.  3.    (2  -  a:)4. 

4.    (2z  +  3y)3.  5.    (3x  -  4y)3.  6.    (3o  +  x2)6. 

7.    (x*+  y*)*.  8.    (or1  +  2ay~1)*.          9.    (a"1  -  x~2)*. 


280  MATHEMATICS  [XVI,  §  212 


10.  (a2  -  b2)8. 

11.  (3a2b  +  2C3)8. 

12.  (1  +  x)10. 

(<!•             9  V0 
l+f)  . 

14.  (2x  -  £)9. 

15.  (5  +  i)8- 

16.  (4.9)3. 

17.  (1.01)B. 

18.  (0.99)«. 

19.  (1.9)5. 

20.   (1.02)4. 

21.  (15/8)7. 

22.  Expand  (1  +  i)5  and  (2  -  f)5  and  check  results. 

23.  Prove  that  any  binomial  coefficient,   counted  from  the  first,  is 

equal  to  the  same  numbered  one,  counted  from  the  last. 

212.  Selected  Terms.  To  select  a  particular  term  in  the 
expansion  of  a  binomial  without  computing  the  preceding  terms, 
we  can  use  the  formula  for  the  (r  +  l)th  term,  namely, 

the  first  r  terms  of  n ! 

nCrxn^ryr  =  -  xn^yr. 

r\ 

(x          \20 
9  ~~  2y  )     . 

Here  r  +  1  =  10,  r  =  9,  n  =  20,  and  the  required  term  is 

20- 19- 18- 17- 16- 15- 14- 13- 12 /x\"  _    _  41990xlly9 

9-8-7-6.5-4. 3-2-1  \2  )    ( 

(r-        1  V3 
\x  +  -  j     which  contains  x2. 

The  (r  +  l)th  term  is  iSCfr(x1/2)13-r(x-1)r  =  nCr-x«3-3r^2,  whence  r 
must  be  3  and  the  4th  term  is  required.  It  is 


EXERCISES 

1.  Find  the  4th  term  of  (4a  -  6)12. 

2.  Find  the  llth  term  of  (2x  -  y)17. 

3.  Find  the  6th  term  of  (xVy  +  yVx)9. 

4.  Find  the  middle  term  of  (x  +  3?/)8. 

(x      2  V° 
-  +  -  j     which  does  not  contain  x. 

fx          y2\12 

6.  Find  the  term  of  I *s.  I    which  contains  neither  x  nor 


XVI,  §213]         THE   BINOMIAL  EXPANSION  281 

213.  The  Binomial  Series.  The  binomial  theorem  and  the 
symbols  nCr  for  the  number  of  combinations  of  n  things  taken  r 
at  a  time,  have  no  meaning  except  when  n  and  r  are  positive 
integers.  On  the  other  hand  we  know  that  such  expressions  as 

(1  +  i)*/2,     (2  +  5)-2,     (32  +  3)1/*,     (1  -  O.I)-1/2, 

have  perfectly  definite  meanings;  e.  g.,  (2  +  5)~2  =  1/49. 

If  we  should  expand  a  binomial  whose  exponent  is  not  a 
positive  integer  by  the  binomial  theorem  (that  is  form  the 
coefficients  and  exponents  by  the  same  rules  as  though  the 
exponent  were  a  positive  integer),  we  should  get  a  non-termi- 
nating series  of  terms.  For  example, 


(32  +  3)1/5  =  321/5 

+  Tfy(32)-""(3)«  ----  . 

Now  it  is  shown  in  advanced  courses  in  mathematics,  that 
this  binomial  series  is  actually  valid,  provided  the  numerical  value 
of  the  first  term  of  the  binomial  is  greater  than  the  numerical  value 
of  the  second  term.  It  is  then  valid,  in  the  sense  that  if  we 
begin  at  the  first  and  add  term  after  term,  the  more  terms  we 
take  the  nearer  the  sum  approaches  to  the  true  value  sought 
and  that,  by  taking  terms  enough,  the  sum  which  we  are  com- 
puting will  approximate  the  true  value  as  nearly  as  we  please. 

EXAMPLE.     Find  VlO  by  the  binomial  series. 
VlO  =  (8  +  2)1/3  =  2(1  +  i)1/3 


Whence  computing,  we  have 


+ 

1.0000-  •  • 
.0833-  •  • 
.0010-  •  • 
.0000-  •  • 

0.0069-  •  • 
.0002-  •• 
.0000-  •• 

1.0843--- 
.0071  •  •  • 

1.0772--- 
2 

1.0843--- 

0.0071 

2.1544-  •• 

=  VIo. 


282  MATHEMATICS  [XVI,  §  214 

The  student  should  note  carefully  that  while  the  binomial 
series  for  (1  +  £)1/3  is  valid,  that  for  (-£  +  1)1/3  is  not. 

EXERCISES 

Expand  the  following  in  binomial  series  and  simplify  five  terms. 

1     (1  4-  x)1^2  2     (1  -r-  x)"1/2  3     (1  x)"1/3 

4.  (0.98)1/3.          5.  (1.02)1/2.  6.  (0.99)1/2. 

7.  V96.      8.  v/30.       9.  v'tKJ.      10.  -\/33~. 
11.  \/15.  12.  v/65.  13.  \/732. 


14. 

V1025. 

15.    ^2400. 

16 

.    ^125. 

17. 

flVimv  thnt  , 

11       1  ,«2      1 

1  .  3^ 

,4    ,    1 

•  3- 

5 

**  + 

... 

T   2*     ~T 

2-  4 

'  2 

•  4- 

6 

18. 

01,                            1 

_1_    1       _L 

1  •  4   . 

'  1  x 

•  4- 

7 

C3  + 

... 

VI   -  x 

1    +   3X   ~T 

3-  6 

1  3 

•  6- 

9 

19. 

Show  that  (1  —a;)"2 

=  1  +  2x  + 

3x2  + 

4x3  -i 

..... 

20. 

Ohow  that       1 

-1       *x+- 

•  3r2 

1- 

3-  £ 

i« 

'  +  ' 

"•. 

Vl  +x 

214.  Mendel's  Law.*  An  Austrian  monk  by  the  name  of 
Mendel  planted  some  sweet  peas  of  different  colors  in  the  garden 
of  the  monastery.  These  blossomed  and  produced  seed.  This 
seed  was  gathered  and  planted  the  following  year.  The  flowers 
produced  the  second  summer  contained  all  of  the  colors  of  the 
first  summer,  but  other  colors  were  present.  By  observing  and 
counting  the  number  of  flowers  of  each  color  Mendel  discovered 
the  law  which  bears  his  name.  In  its  simplest  form  it  may  be 
explained  as  follows. 

Suppose  a  bed  of  sweet  peas  with  blossoms  half  of  which  are 
red  and  half  of  which  are  white.  Fertilization  of  the  flowers 
by  wind  and  insects  will  take  place  without  selection.  That  is, 
pollen  from  a  white  flower  is  equally  likely  to  fertilize  a  red  or  a 
white  flower.  If  pollen  from  a  white  flower  fertilizes  a  white 

*  The  following  articles  (§§  214-216)  are  based  largely  upon  Chapter  XIV  of  E. 
DAVENPORT,  Principles  of  Breeding,  Ginn  and  Co.  Much  additional  information  may 
be  found  there. 


XVI,  §  215] 


LAWS  OF  HEREDITY 


283 


flower  the  seed  produced  is  of  pure  stock  and  will  produce  pure 
white  flowers  the  following  year.  Such  flowers  let  us  denote  by 
W2.  If  pollen  from  a  white  flower  fertilizes  a  red  flower,  or  vice 
versa,  the  seed  produced  will  be  mixed  stock  and  the  following 
year  will  show  its  mixed  character  by  producing  flowers  which  are 
neither  red  nor  white  but  some  intermediate  shade.  Such 
flowers  let  us  denote  by  RW.  The  symbol  R2  is  now  self-explan- 
atory. On  counting  the  flowers  which  are  pure  white,  mixed, 
and  red,  we  would  discover  their  numbers  to  be  approximately 
in  the  ratio  1:2:  1.  These  are  the  coefficients  in  the  expansion 
of  (R  +  W)2.  This  is  what  one  might  have  expected  beforehand, 
as  is  seen  from  the  adjoined  table.  Observe  that  there  are  twice 
as  many  flowers  of  mixed  color  as  of  either  of  the  pure  colors. 


Color  of  fertilizing  flower. 

Color  of  flower  fertilized. 

R 

W 

R... 

R? 
RW 

RW 

W2 

w  

Result  of  mixing :  R2  +  2RW  +  W2. 

215.  Successive  Generations.  Let  R*  denote  the  result  of 
fertilizing  R2  with  R2 ;  RSW  denote  the  result  of  fertilizing  R2 
with  RW,  and  so  on.  Then  the  results  of  indiscriminate  fertili- 
zation of  the  flowers  will  be  shown  in  the  second  generation,  but 
in  the  third  year,  as  given  in  the  following  table. 


Color    of    fertilizing 
flower  and  its  relative 
numbers. 

Color  of  flower  fertilized  and  their  relative  numbers. 

fi» 

2RW 

W* 

&.. 

R4 
2R3W 
R*W* 

2R3W 
4R?W2 
2RW3 

/PTP 
2RW3 
W* 

2RW  

JP  

Result  of  mixing :    R*  +  4RW  +  6RW2  +  4RW3  +  W* 
Observe  that  the  result  in  the  second  generation  of  mixing  is 
the  binomial  expansion  of  (R  +  W)4. 


284  MATHEMATICS  [XVI,  §216 

Similarly  we  can  show  that  the  result  in  the  third  generation 
of  mixing  is  given  by  (R  +  W)8,  and  so  on. 

216.  Mixing  of  Three  Colors.  Make  a  table,  as  above,  but 
for  three  colors.  Suppose  the  third  color  to  be  blue  (B).  Then 
a  complete  expression  for  the  effect,  in  the  first  generation  after 
mixing,  is  the  following : 

(R  +  W  +  BY  =  R2  +  W2  +  B2  +  2RW  +  2RB  +  2WB. 

In  case  the  ratio  of  the  number  of  white  flowers  to  red  flowers 
is  as  2  to  3  then  the  result  in  the  first  generation  after  mixing 
is  as  follows : 

(2W  +  3fl)2  =  4PF2  +  12WR  +  9fl2. 

Mendel's  law  of  heredity,  as  illustrated  above  by  the  dis- 
tribution of  color  in  the  successive  generations  of  plants,  applies 
to  other  transmissible  characters  in  both  plants  and  animals. 
That  this  distribution  follows  the  mathematical  laws  of  the 
binomial  formula  is  due  to  the  fact  that  each  individual  plant 
or  animal  inherits  the  characteristics  of  two  parents,  and  hence 
the  number  two  and  its  mathematical  properties  have  their 
analogies  in  the  laws  of  biology. 

EXERCISES 

1.  Plot  a  few  graphs,  using  binomial  coefficients  as  ordinates  and 
the  number  of  the  corresponding  term  as  abscissas. 

2.  How  many  varieties  of  sweet  peas  are  produced  by  sowing  in  the 
same  bed  three  different  strains  (a)  first  year ;    (6)  second  year. 

Ans.    (a)  6;  (6)  14. 

3.  A  farmer  buys  two  different  kinds  of  thoroughbred  chickens  but 
allows  them  to  mix  freely.     How  many  different  kinds  of  chickens  will 
he  have  at  the  end  of  (a)  the  first,  (6)  the  second,  (c)  the  third  year  of 
hatching?  Ans.    (a)  3,    (6)  5,  (c)  9. 

4.  Four  different  varieties  of  wheat  are  planted  side  by  side.     How 
many  different  varieties  will  be  harvested?  Ans.    10. 

5.  Plot  graphs  as  indicated  in  Ex.  1  for  the  results  of  Ex.  3. 


XVI,  §  216]  LAWS  OF  HEREDITY  285 

6.  What  varieties  and  in  what  proportion  are  obtained  by  freely 
mixing  the  first  and  second  generations? 

7.  I  plant  8  sweet  pea  seeds  —  4  red,  4  white.     Each  seed  produces 
16  flowers  —  each  flower  matures  2  seeds  which  germinate  and   grow 
the  following  season.     Find  the  total  number  of  flowers,  the  proportion 
and  number  of  the  different  kinds  of  flowers,  in  the  (a)  first,  (6)  second, 
and  (c)  third  generations. 


CHAPTER  XVII 
THE  COMPOUND   INTEREST  LAW 

217.  Compound  Interest.  Suppose  one  dollar  to  be  loaned 
at  compound  interest  at  r%  per  annum  payable  annually.  The 
interest  i,  due  at  the  end  of  the  first  year,  is  r/100.  The  amount 
due  is  1  +  i-  If  interest  is  payable  semiannually  the  amount 
due  at  the  end  of  the  first  half  year  is  1  +  i/2*  If  the  interest 
is  payable  quarterly  the  amount  due  at  the  end  of  the  first  quar- 
ter is  1  +  i/4. 

In  general  terms  if  the  interest  is  payable  p  times  a  year  at 
r%  per  annum  compound,  the  amounts  due  on  a  principal  of 
one  dollar  at  the  end  of  the  1st,  2d,  •  •  •,  pth  period  are  respec- 
tively, 


and  the  amounts  due  at  the  end  of  the  1st,  2d,  •••,  nth  years  are 
respectively, 

(i\p 
1+*) 
pJ    \      p         \     p 

The  amount  A  at  the  end  of  n  years  at  r%  per  annum  payable 
p  times  a  year  on  a  principal  of  P  dollars  is  given  by  the  formula 

*  The  amount  of  one  dollar  for  n  years  compound  interest  at  r%  payable  annually 
is  (1  +  i)n.  If  a  settlement  is  made  between  two  interest  dates  there  is  some  divergence 
of  practice  in  computing  the  interest  for  the  fractional  part  of  a  year.  The  amount  of 
one  dollar  for  the  pth  part  of  a  year  by  analogy  to  (1  +  t)w  would  be  (1  +t)  l/p  =  Vl  +  i, 
but  1  +  -  is  often  used  instead.  When,  however,  by  the  terms  of  the  note  the  interest 
is  payable  p  times  a  year,  and  is  to  be  compounded,  it  is  clear  that  the  amounts  due  at 
the  end  of  1,  2,  •••,  n  periods  are 


+LY  .., 

p/ 


286 


XVII,  §  218]         THE  COMPOUND  INTEREST  LAW  287 


(,)  A  -  P( 


218.  Continuous  Compounding.  The  larger  p  is  the  shorter 
the  interval  between  the  successive  interest  paying  dates.  As 
p  increases  without  bound  this  interval  approaches  zero  ;  i.e. 
we  can  take  p  large  enough  to  make  this  interval  as  small  as  we 
please.  In  the  limit  interest  is  said  to  be  compounded  contin- 
uously. While  this  state  is  never  realized  in  financial  affairs 
it  is  closely  approximated.  For  example,  large  retail  stores 
sell  goods  over  the  counter  very  nearly  continuously  and  con- 
tinuously replenish  their  stock. 

Let  us  see  what  form  equation  (1)  takes  when  p  becomes 
infinite.  Put  x  for  i/p  which  approaches  zero  when  p  becomes 
infinite.  Then  (1)  becomes 

in  1 

(2)  A  =  P(l  +  x}*  =  P[(l  +  x)x]in. 

Now  it  is  shown  in  books  on  the  Calculus  that  as  x  approaches 

i 

zero,  the  quantity  (1  +  x)x  converges  to  a  certain  number  be- 
tween 2  and  3.  This  number  is  the  base  of  the  natural  or 
Napierian  system  of  logarithms  and  is  usually  denoted  by  e. 
To  five  decimal  places  c  =  2.71828.  It  can  be  shown  that  the 
following  steps  are  justifiable,  although  the  proof  will  not  be 
given  here.  By  the  Binomial  Formula, 


x 


As  x  approaches  zero  the  terms  on  the  right  converge  respectively 
to  the  terms  of  the  series 


288  MATHEMATICS  [XVII,  J  218 

If  we  begin  at  the  first  and  add  the  terms  of  this  series,  the 
more  terms  we  add  the  nearer  the  sum  comes  to  e.     „  00000  0 

The  sum  of  the  first  ten  terms  is  2.71828,  as  is  shown 

u.ouuuu  u 

7 
7 


in  the  adjoining  computation.  ~ 

Then   we   conclude   that   as   x   approaches   zero 


i 


0.04166 


(1  +  x)x  converges  to  e. 

Returning  now  to  equation  (2)  we  see  that  as  p  0.00138  9 

becomes  infinite  and  x  approaches  zero,  A  converges  0.00019  9 

to  Pein.     Hence  we  say  that  when  interest  is  com-  0.00002 

pounded  continuously,  the  amount  of  P  dollars  at  0-00000  2 


r%  per  annum  for  n  years  is  given  by  the  equation     2.71828 
(3)  A  =  Pein, 

in  which  i  =  r/100  is  the  simple  interest  on  one  dollar  for  one 
year.  This  equation  is  said  to  represent  the  compound  interest 
law. 

Scientific  investigations  reveal  many  examples  of  quantities 
whose  rate  of  increase  (or  decrease)  varies  as  the  magnitude  of 
the  quantity  itself.  For  example,  the  number  of  bacteria  in  a 
favorable  medium,  or  the  growth  of  an  organic  body  by  cell 
multiplication ;  again  the  rate  of  decrease  in  atmospheric  pres- 
sure in  ascending  a  mountain  is  proportional  to  the  pressure, 
and  the  rate  of  change  in  the  volume  of  a  gas  expanding  against 
resistance  varies  as  the  volume.  The  proverbial  phrases,  the 
rich  grow  richer,  the  poor  poorer;  nothing  succeeds  like  success; 
a  stitch  in  time  saves  nine;  are  expressions  in  popular  language 
which  show  a  recognition  of  this  law  in  crude  form.* 

In  general  terms  if  y  and  x  are  two  varying  quantities  such 
that  the  rate  of  change  in  y  (as  regards  a  change  in  x)  is  known  to 
vary  directly  as  y  itself,  then  they  are  connected  by  an  equation 
of  the  form 

*  See  DAVIS,   The  Calculus,  §  81. 


XVII,  §  218]        THE  COMPOUND  INTEREST  LAW  289 


(4)  y  = 

in  which  c  and  k  are  constants. 

EXAMPLE.  Suppose  that  atmospheric  pressure  at  the  earth's  surface 
is  15  Ibs.  per  square  inch  and  that  it  is  10  Ibs.  per  square  inch  at  a  height 
of  12,000  ft.  If  now  it  be  assumed  that  the  rate  of  decrease  in  the  pres- 
sure is  proportional  to  the  pressure,  we  have  from  equation  (4) 

p  =  cetrt. 

Substituting  p  =  15  when  &  =  0,   we  find  c  =  15;    then   substituting 
p  =  10,  h  =  12,000,  c  =  15,  we  find 


12000 ' 


and  these  values  of  c  and  k  give 


12000 


p  = 

by  means  of  which  the  pressure  at  any  height  h  can  be  computed. 

This  example  illustrates  the  method  of  solving  similar  problems  which 
fall  under  the  compound  interest  law.  We  assume  an  equation  of  the 
form  of  (4)  and  determine  the  constants  c  and  k  by  substituting  in  known 
pairs  of  values  of  x  and  y.  Having  determined  the  constants  we  insert 
them  in  the  assumed  formula  which  is  then  in  form  to  give  the  value 
of  y  corresponding  to  any  value  whatever  of  x. 

EXERCISES 

1.  Do  you  see  any  relation  between  the  growth  of  plants,  or  the 
increase  in  population,  and  the  compound  interest  law?     Is  the  relation 
exact?     What  circumstances  tend  to  limit  its  application? 

2.  Is  there  any  relation  between  your  ability  to  acquire  knowledge 
and  to  think  clearly  and  the  compound  interest  law? 

3.  The  population  of  the  state  of  Washington  was  349,400  in  1890  and 
in   1900  it  was  518,100.      Assume  the  relation  P  =  ceT,  where  P  = 
population,  T  =  time  in  years  after  1890,  and  predict  the  population 
for  1910. 

4.  Using  the  data  of  Ex.  3,  find  the  average  annual  rate  of  increase 
from  1890  to  1900.     Assuming  the  same  average  rate  to  be  maintained 
for  the  next  10  years,  predict  the  population  for  19^0. 


290  MATHEMATICS  [XVII,  §  218 

5.  When  heated,  a  metal  rod  increases  in  length  according  to  the 
compound  interest  law.     If  a  rod  is  40  ft.  long  at  0°  C.,  and  40.8  ft.  long 
at  100°  C.,  find  (a)  its  length  at  300°  C;   (&)  at  what. temperature  its 
length  will  be  41  ft.  6  in.  Ans.    (a)  42.448 ;  (&)  185°.8 

6.  The  rate  of  increase  in  the  tension  of  a  belt  is  proportional  to  the 
tension  as  the  distance  changes  from  the  point  where  the  belt  leaves  the 
driven  pulley.     If  the  tension  =  24  Ibs.  at  the  driven  pulley,  and  32  Ibs. 
ten  feet  away,  what  is  it  six  feet  away?  fAns.   28.52 

7.  Assuming  that  the  rate  of  increase  in  the  number  of  bacteria  in  a 
given  quantity  of  milk  varies  as  the  number  present,  if  there  are  10,000 
at  6  A.M.,  60,000  at  9  A.M.,  how  many  will  there  be  at  2  P.M.?     At 
3  P.M.?    At  6  P.M.?  Ans.   2  P.M.,  1,188,700. 

8.  In  the  process  of  inversion  of  raw  sugar,  the  rate  of  change  is  pro- 
portional to  the  amount  of  raw  sugar  remaining.     If  after  10  hours  1000 
Ibs.  of  raw  sugar  has  been  reduced  to  800  Ibs.,  how  much  raw  sugar  will 
remain  at  the  end  of  24  hours?  Ans.   586  Ibs. 


CHAPTER  XVIII 
PROBABILITY 

219.  Definition  of  Probability.     //  an  event  can  happen  in  h 
ways,  and  fail  in  f  ways,  the  total  number  of  ways  in  which  the 
event  can  happen  and  fail  is  h  +/.     Then  h/(h  +/)  is  said  to 
be  the  probability  that  the  event  will  happen,  andf/(h-\-f)  is  said 
to  be  the  probability  that  the  event  will  fail. 

For  example,  suppose  we  have  a  box  containing  4  red  marbles 
and  5  white  ones.  Let  us  determine  the  chance  of  drawing  a 
red  marble  the  first  time.  This  event  can  happen  in  4  ways,  and 
fail  in  5  ways,  while  the  total  number  of  ways  in  which  the 
event  can  happen  and  fail  is  nine.  Then  by  the  preceding  defi- 
nition the  probability  of  drawing  a  red  marble  is  4/9,  and  the 
probability  of  not  drawing  a  red  marble  is  5/9.  Observe  that 
one  of  these  things  is  certain  to  happen.  The  measure  of  this 
certainty  is  the  sum  of  the  probabilities  of  the  separate  events.  This 
sum  is  1 .  Hence,  if  p  is  the  probability  that  an  event  will  happen, 
the  probability  q  that  it  will  not  happen  is  1  —  p. 

220.  Statistical  Probability.     In  a  throw  of  a  penny,  before 
the  event  takes  place,  there  is  no  reason  to  suspect  that  heads 
are  more  likely  to  turn  up  than  tails.     In  a  throw  of  a  die  any  one 
of  the  six  faces  is  equally  likely  to  turn  up  and  this  probability 
does  not  depend  upon  the  particular  die  used.     The  probability 
of  a  man's  making  a  safe  hit  in  a  game  of  baseball,  and  that  of 
not  making  a  safe  hit  are  not  equal.     Here  the  individuality 
of  the  batter  enters  and  before  the  event  takes  place,  if  the  batter 

291 


292  MATHEMATICS  [XVIII,  §  220 

is  unknown,  we  have  nothing  on  which  to  make  an  estimate. 
If  the  batter  is  known,  our  estimate  is  based  on  his  past  perform- 
ance and  this,  unlike  a  throw  of  dice,  depends  upon  the  particu- 
lar individual  at  bat.  If  out  of  the  last  60  times  at  bat,  he  has 
made  a  safe  hit  20  times,  then  we  say  that  the  probability  of  his 
making  a  safe  hit  this  time  at  bat  is  1/3. 

Again  what  is  the  probability  that  a  man  aged  70  will  die 
within  the  next  year?  Clearly  this  depends  upon  the  individual, 
his  present  state  of  health,  his  habits,  etc.  In  this  case,  how- 
ever, we  can  construct  a  measure  of  his  probability  of  dying 
which  is  independent  of  these  personal  elements.  From  the 
American  Experience  Mortality  Table  (see  Tables,  p.  329),  we 
find  that  out  of  38,569  persons  living  at  age  70,  within  the  year 
2,391  die.  Hence  the  probability  that  a  man  aged  70  will  die 
within  the  year  is  2,391  -4-  38,569. 

To  derive  the  probability  of  an  event  from  statistical  data  divide 
the  number  of  cases  h  in  which  the  event  happened  by  the  total  num- 
ber n  of  cases  observed. 

221.  Expectation.  If  p  is  the  probability  that  a  man  will 
win  a  certain  sum  s  of  money,  then  the  product  sp  is  called  the 
value  of  his  expectation. 

Thus  the  value  of  a  lottery  ticket  in  which  the  prize  is  $25 
and  in  which  there  are  500  tickets  is  $25  X  1/500,  or  30  cents. 

EXERCISES 

1.  According  to  the  mortality  table   (p.   329)   it  appears  that  of 
100,000  persons  at  the  age  of  10,  only  5,485  reach  the  age  of  85.     What  is 
the  probability  that  a  child  aged  10  will  reach  the  age  of  85? 

2.  On  200  of  240  school  days  a  student  has  had  a  grade  of  90.     What 
is  the  probability  that  his  grade  will  be  90  on  the  241st  day? 

3.  The  weather  bureau  predicts  rain  for  to-day.     What  is  the  prob- 
ability that  it  will  rain,  if  on  the  average  90  out  of  every  100  predictions 
are  correct? 


XVIII,  §  223]  PROBABILITY  293 

/      4.   Compute  the  probability  of  throwing  with  2  dice  a  sum  of  (a) 
seven,  (6)  eight,  (c)  nine,  (d)  ten,  (e)  eleven. 

Ans.    (a)  i;  (6)  &;  (c)  *;  (d)  & ;  (e)  &. 

5.  Find  the  probability  in  drawing  a  card  from  a  pack  that  it  be  (a)  an 
ace,  (6)  a  spade,  (c)  a  face  card,  (d)  not  a  face  card. 

Ans.    (a)  A;  (6)  i;    (c)  &;    (d)  £. 

6.  Find  the  expectation  of  a  man  who  is  to  win  $300  if  he  holds  one 
ticket  out  of  a  total  of  1000  tickets.  Ans.   30  cents. 

222.  Mutually  Exclusive  Events .     Two  events  are  said  to  be 
mutually  exclusive  if  the  occurrence  of  one  of  them  precludes  the 
occurrence  of  the  other.     For  example,  in  a  race  between  A,  B, 
and  C,  if  A  wins,  B  and  C  do  not  win. 

//  the  probabilities  of  the  mutually  exclusive  events  E\,  E%,  •••, 
En  are  p\,  pz,  •••,  pn,  then  the  probability  that  some  one  will  occur 
is  the  sum  of  the  probabilities  of  the  separate  events. 

The  meaning  will  be  made  clear  by  means  of  the  following 
illustration.  A  bag  contains  3  red,  4  white,  and  5  blue  balls. 
What  is  the  probability  that  in  a  first  draw  we  obtain  a  red  or 
a  white  ball?  There  are  12  balls  in  all  and  7  cases  are  favorable, 
namely  3  red  and  4  white  balls.  Then  from  the  definition  of 
probability  the  chance  of  drawing  a  red  ball  or  a  white  ball  is 
7/12.  But  the  probability  of  drawing  a  red  ball  is  3/12  and  that 
of  drawing  a  white  ball  is  4/12  and  (3/12)  +  (4/12)  =  7/12. 

223.  Dependent  Events.     Events  are  said  to  be  dependent 
if  the  occurrence  of  one  influences  the  occurrence  of  the  other. 
//  the  probability  of  a  first  event  is  p\ ;  and  if  after  this  has  happened 
the  probability  of  a  second  event  is  p^;    etc.,  ••• ;    and  if  after  all 
those  have  happened  the  probability  of  an  nth  event  is  pn ;   then  'the 
probability  that  all  of  the  events  will  happen  in  the  given  order  is 
Pi,  P2  "•  pn. 

For,  if  the  first  event  can  happen  in  hi  ways  and  can  fail  in  /i 
ways ;  and  if  after  this  has  happened  the  second  can  happen  in  h% 
ways  and  can  fail  in  fa  ways ;  etc.,  ••• ;  and  if  after  these  have  hap- 


294  MATHEMATICS  [XVIII,  §  223 

pened  the  nth  event  can  happen  in  hn  ways  and  can  fail  in  /„ 
ways  ;  then  they  can  all  happen  and  fail  in  (hi  +  fi)(hz  +  f%)  ••- 
(hn  +  /»)  ways.  Now  all  the  events  can  happen  together  in  the 
given  order  in  hi  A2  •••  hn  ways.  Then  by  the  definition  of  prob- 
ability the  chance  that  all  of  the  dependent  events  will  take 
place  in  the  given  order  is 

hi  hz  "-  hn  _       h\  hi  hn 

'  " 


(hi  +fi)(h2  +/2)  -.  (hn  +/„)      hi  +/!      A2+/2  +/„ 

=   PlP*    '"  Pn- 

Thus  the  problem  of  drawing  2  red  balls  in  succession  from  a 
bag  containing  3  red  and  2  black  balls  is  (3/5)  X  (2/4)  =  3/10. 
For  after  drawing  one  red  ball  and  not  replacing  it  the  probability 
of  drawing  a  red  ball  the  second  time  is  2/4. 

224.  Independent  Events.  Events  are  said  to  be  independent 
when  the  occurrence  of  any  one  of  them  has  nothing  to  do  with 
the  occurrence  of  the  others. 

The  probability  that  all  of  a  set  of  independent  events  will  take  place 
is  the  product  of  the  probabilities  of  the  independent  simple  events. 

This  follows  as  a  corollary  from  the  theorem  of  §  223. 

Thus  the  probability  of  throwing  a  deuce  twice  in  succession 
is  (1/6)  X  (1/6)  =  1/36. 

EXERCISES 

1.  If  the  batting  average  of  Tyrus  Cobb  is  0,400  what  is  the  chance 
that  in  any  single  time  at  bat  he  will  make  a  safe  hit  ? 

2.  What  is  the  probability  of  holding  4  aces  in  a  game  of  whist? 

Ans.    1/270,725. 

3.  Suppose  I  enter  2  horses  for  a  race  and  that  the  probabilities  of 
their  winning  are  respectively  |  and  j.     What  is  the  probability  that 
one  or  the  other  will  win  the  race?  Ans.    3/4. 

4.  Does  Ex.  3  teach  us  anything  with  respect  to  diversified  farming? 
Discuss  the  probability  of  crop  failure  of  a  single  crop  as  compared  with 
that  of  two  or  more  different  crops. 

5.  Three  men  A,  B,  C  go  duck  hunting.     A  has  a  record  of  one  bird 


XVIII,  §  223]  PROBABILITY  295 

out  of  two,  B  gets  two  out  of  three,  C  gets  three  out  of  four.     What  is 
the  probability  that  they  kill  a  duck  at  which  all  shoot  at  once  ? 

Ans.   23/24. 

6.  What  is  the  chance  of  drawing  a  white  and  red  ball  in  the  order 
named  from  a  bag  containing  5  white  and  6  red  balls?      Ans.   3/11. 

7.  In  a  certain  zone  in  times  of  war  23  out  of  5000  ships  are  sunk  by 
submarine  in  one  week.     What  is  the  chance  that  a  single  vessel  will 
cross  the  zone  safely?     What  is  the  chance  that  all  of  4  vessels  which 
enter  the  zone  at  the  same  time  will  cross  in  safety  ?     What  is  the  chance 
that  of  these  4  exactly  3  will  cross  in  safety  ?     That  at  least  3  will  cross 
in  safety? 

8.  In  certain  branches  of  the  army  service  2%  of  the  men  are  killed 
each  year.     Three  brothers  enlist  in  this  branch  of  the  service  for  a 
period  of  two  years.     Compute  the  probability  that  (a)  all  will  survive, 
(b)  exactly  2  will  survive,  (c)  at  least  2  will  survive,  (d)  exactly  one  will 
survive,  (e)  at  least  one  will  survive,  (/)  none  will  survive. 

9.  At  the  time  of  marriage  the  probabilities  that  a  husband  and  wife 
will  each  live  50  years  are  \  and  j  respectively.     Compute  the  probabil- 
ity that  (a)  both  will  be  alive,  (b)  both  dead,  (c)  husband  alive  and  wife 
dead,  (d)  wife  alive  husband  dead. 

10.  From  the  American  Experience  Table  of  Mortality   (Tables, 
p.  329)  compute  your  chances  of  living  1,  10,  20,  30,  40,  50  years. 

11.  From  the  American  Experience  Table  of   Mortality  (Tables, 
p.  329)  find  that  age  to  which  you  now  have  an  even  chance  of  living. 

12.  Find  from  the  same  table  that  age  to  which  a  person  aged  20  has 
an  even  chance  of  living.  Ans.   66+. 

13.  Three  horses  are  entered  for  a  race.     The  published  odds  are  5  :  4 
for  A ;   3:2  against  B ;  4:3  against  C.     Is  it  possible  to  place  bets  in 
such  a  way  that  I  win  some  money  no  matter  which  horse  wins  ? 

Ans.   Yes. 

14.  Suppose  n  horses  entered  for  a  race,  and  let  the  published  odds 
be  (a  —  1)  to  1  against  the  first ;  (6  —  1)  to  1  against  the  second,  (c  —  1) 
to  1  against  the  third  and  so  on.     A  man  bets  (a  —  l)/a  to  I/a  against 
the  first;  (b  —  l)/6  to  1/6  against  the  second,  etc.     Show  that  whatever 
horse  wins  his  gains  are  represented  algebraically  by  the  formula 

f5+;  + 


296 


MATHEMATICS 


[XVIII,  §  225 


225.  Frequency  Distribution  Curves.*  A  sample  of  400  oats 
plants  were  taken  from  an  experimental  plot  and  measured  as  to 
height  in  centimeters  with  the  following  results  :  f 


Height,  H   

45 
50 

50 
55 

55 
60 

60 
65 

65 
70 

70 
75 

75 

80 

80 
85 

85 
90 

90 
95 

Frequencies,  F  

2 

9 

21 

34 

97 

m 

89 

?4 

0 

1 

Let  us  plot  this  data  with  heights  as  abscissas  and  frequencies 
as  ordinates.     Construct  rectangles,  with  bases  on  the  horizontal 


rr1 

^, 

/ 

\ 

/ 

v 

\ 

/ 

80 

/ 
/ 

V 

/ 

- 

\ 
\ 

-GO 

\ 

/ 

j 

—10 

/ 
/ 

\ 
\ 

i 
j 

/ 

' 

/ 

^ 

.- 

"" 

\ 
5 

s  i 

'i 

45 

5 

0 

5 

5 

6 

0 

f 

5 

7 

0 

' 

5 

E 

0 

f 

0 

<J 

0 

9 

5 

/*• 

FIG.  128 

axis.     Let  the  width  of  the  base  in  each  case  be  5  units,  which 
agrees  with  the  grouping  of  the  measurements  as  to  height. 

*  In  the  remainder  of  this  Chapter  (§§  225-231),  the  authors  are  indebted  for  many 
ideas  to  E.  DAVENPORT,  Principles  of  Breeding  [Chapter  XII  and  Appendix  (H.  L. 
HIETZ)].  Other  books  containing  similar  matter  are  JOHNSON,  Theory  of  Errors  and 
Method  of  Least  Squares:  WRIGHT  AND  HAYFORD,  Adjustments  of  Observations;  MERRI- 
MAN,  Textbook  of  Least  Squares;  WELD,  Theory  of  Errors  and  Least  Sqiiares;  etc. 

t  MEMOIR  No.  3,  CORNELL  UNIVERSITY  AGRICULTURAL  EXPERIMENT  STATION, 
Variation  and  Correlation  of  Oats,  by  H.  H.  LOVE  and  C.  E.  LEIGHTY,  Aug.,  1914. 


XVIII,  §  226]  PROBABILITY  297 

Let  the  height  of  the  individual  rectangles  be  representative  of 
the  frequency  for  the  corresponding  heights  of  plants,  as  shown 
in  Fig.  128. 

The  upper  parts  of  these  rectangles  form  an  irregular  curve 
made  up  of  segments  of  straight  lines.  A  smoother  curve  is 
obtained  by  connecting  the  middle  points  of  the  upper  bases  of 
these  rectangles  by  segments  of  straight  lines  as  shown  by  the 
dotted  line  in  Fig.  128.  Instead  of  the  dotted  line  we  may  draw 
a  smooth  curve  as  near  as  possible  to  the  middle  points  of  the 
upper  bases.  Any  curve  drawn  as  nearly  as  possible  through 
a  series  of  plotted  points  representing  a  distribution  with  respect 
to  a  given  character  is  called  a  frequency  distribution  curve. 

Such  curves  are  useful  in  presenting  to  the  eye  some  of  the 
features  of  a  distribution.  The  type  of  character  most  fre- 
quent is  represented  by  the  mode  (§  198),  which  is  the  value 
of  the  abscissa  corresponding  to  the  highest  point  of  the  curve. 
The  median  measurement  of  the  group  (§  197)  is  represented  by 
the  abscissa  of  that  ordinate  on  either  side  of  which  there  are 
equal  areas  under  the  curve.  The  arithmetic  average  (§  195) 
is  the  abscissa  of  the  center  of  gravity  of  the  area  under  the 
curve. 

Frequency  distribution  curves  are  plotted  for  a  great  variety 
of  things,  such  as  frequency  distribution  of  people  with  respect 
to  height,  weight,  or  age ;  grains  of  wheat  with  respect  to  weight ; 
alfalfa  with  respect  to  duration  of  bloom  in  days ;  cherry  trees 
with  respect  to  earliness  of  bloom ;  pigs  with  respect  to  size  of 
litter;  diphtheria  with  respect  to  time  of  year;  women  with 
respect  to  age  of  marriage ;  etc. 

226.  Probability  Curve.  If  a  large  number  of  measurements 
are  made  upon  the  same  item,  they  will  not  in  general  agree. 
Let  us  plot  as  abscissas  the  measurements  observed  and  as  ordi- 
nates  their  relative  frequencies.  In  most  cases,  the  positive 


298 


MATHEMATICS 


[XVIII,  §  226 


and  negative  errors  are  equally  likely  to  occur,  and  small  errors 
are  more  numerous  than  large  ones.  The  frequency  curve  for 
the  observed  data  would  then  have  its  highest  point  at  the  true 
value  of  the  measured  magnitude,  would  be  symmetric  about  an 
ordinate  through  this  highest  point,  and  would  rapidly  approach 
the  axis  of  abscissas  both  to  the  right  and  left  of  this  maximum 
ordinate.  If  we  take  the  vertical  through  the  highest  point  as  an 
axis  of  y,  then  abscissas  will  represent  errors  of  observation  and 
ordinates  will  represent  frequency  of  error. 

The  curve  so  drawn  is  well  represented  by  the  equation 


(1) 


y  = 


in  which  cr  is  what  we  shall  call  the  standard  deviation,  e  = 
2.71828  •••  the  base  of  Napierian  logarithms,  n  the  number  of 
observations,  x  the  error  of  a  reading,  y  the  probability  of  an 


error  x.  This  curve  is  called  the  probability  curve  or  curve 
of  error. 

While  the  theoretical  curve  (1)  is  symmetric,  the  curves  ob- 
tained by  plotting  the  results  of  statistical  study  are  often 
not  symmetric.  However  the  formulas  developed  in  this  chapter 
for  the  symmetric  case  can  be  used  for  approximate  results  in  the 
non-symmetric  cases. 

227.   Standard  D  e viation .    It  is  not  enough  to  know  the  value 


XVIII,  §  227]  PROBABILITY  299 

of  the  arithmetic  average  or  the  mode.  It  is  important  to  have 
a  measure  of  the  tendency  to  deviate  from  the  average  or  from 
the  mode. 

The  general  theory  will  be  explained  by  means  of  the  data  of 
§  225,  which  represents  the  measurements  of  the  heights  of  400 
oat  plants.  From  this  data  the  average  height  of  oat  plant  is 
70.8  centimeters.  Compute  the  deviation,  D,  of  these  plants 
from  their  average  height.  Multiply  the  square  of  each  devia- 
tion by  its  corresponding  frequency  and  add  the  results.  We 
get  19,320.  Divide  by  the  sum,  400,  of  the  frequencies.  The 
quotient  is  48.3.  We  next  extract  the  square  root  since  the  de- 
viations have  all  been  squared  in  the  above  calculations.  We 
get  6.95~,  and  this  is  called  the  standard  deviation. 

In  general,  to  find  the  standard  deviation, 

Compute  the  deviation  of  each  frequency  from  the  arithmetic 
average.  Multiply  the  square  of  each  deviation  by  its  corresponding 
frequency  and  add  the  results.  Divide  by  the  sum  of  thefrequenci.es. 
Extract  the  square  root. 

This  rule  is  symbolized  in  the  following  formula : 


(2)  „ 

The  curve  A  in  Fig.  129  represents  the  distribution  when  a  is 
small,  and  the  curve  B  represents  the  distribution  when  a  is  large. 

For  example,  the  two  sets  of  numbers  7,  7,  8,  8,  8,  8,  9,  9  and 
5,  6,  7,  8,  8,  9,  10,  11  have  the  same  arithmetic  mean.  The 
second  set,  however,  shows  a  greater  tendency  to  vary  from 
the  arithmetic  average  (type)  than  does  the  first.  This  greater 
tendency  to  vary  is  shown  by  the  larger  value  for  cr  for  the 
second  set.  The  values  of  a-  are  0.706  and  1.87  respectively. 

Again,  suppose  two  men  are  shooting  at  a  mark,  and  that  we 
compute  the  standard  deviation  for  each.  The  man  for  whom  <r 
is  smallest  is  said  to  be  the  more  consistent  shot. 


300  MATHEMATICS  [XVIII,  §  228 

228.  Coefficient  of  Variability.  A  comparison  of  the  standard 
deviations  of  two  different  groups  conveys  little  information 
as  to  their  respective  tendencies  to  deviate  from  the  arithmetic 
average.     This  is  due  to  two  causes :  (1)  the  measurements  may 
be  in  different  units,  as  centimeters  and  grams,  (2)  one  average 
may  be  much  larger  than  the  other,  for  example  the  average 
height  of  a  group  of  men  would  be  larger  than  the  average  length 
of  ears  of  corn.     We  need  then  a  measure  of  variability  which 
is  independent  of  the  units  used  and  takes  into  account  the 
relative   magnitudes   of   the   means.     Such  a   measure   is   the 
coefficient  of  variability,  which  is  denoted  by  C  and  is  determined 
by  the  formula, 

/Ox  r  _  Standard  deviation  _  <r 

\y)  v  —  "7 — r~j : • 

Arithmetic  average      ra 

For  example,  the  coefficient  of  variability  in  height  of  the  400 
oat  plants  considered  in  §  225  is  6.95/70.8,  or  approximately 
10%. 

229.  Probable  Error  of  a  Single  Measurement.    Any  indi- 
vidual measurement  is  likely  to  be  in  error.     This  error  is  ap- 
proximately the  difference  between  this  measurement  and  the 
arithmetic  average  of  all  the  measurements.     Compute  these 
errors  for  all  the  measurements,  some  positive,  some  negative. 
Give  them  all  positive  signs  and  arrange  them  in  order  of  magni- 
tude.    The  median  of  this  list  is  called  the  probable  error  of  a 
single  measurement  of  the  set  and  is  denoted  by  Es.     It  is 
shown  in  the  theory  of  probability  that 

(4)  Es  =  0.67450-. 

230.  Probable  Error  in  the  Arithmetic  Average.    Take  a 
sample  of  500  ears  of  corn  from  a  crib.     Compute  the  arithmetic 
average  of  their  lengths.     We  use  this  to  represent  the  mean 
length  of  all  the  ears  in  the  crib.     Quite  likely  it  differs  from  their 
true  arithmetic  average.     We  now  find  by  means  of  equation 


XVIII,  §231]  PROBABILITY  301 

(5)  below,  a  number  Em,  called  the  probable  error  in  the  arith- 
metic average.  This  is  a  number  such  that  it  is  equally  likely 
whether  or  not  the  computed  arithmetic  average  of  the  500  ears 
selected  lies  between  ra  —  Em  and  m  +  Em,  where  m  denotes  the 
(unknown)  true  arithmetic  average  for  all  the  ears  in  the  crib. 
In  other  words  if  a  very  large  number  of  persons  take  a  sample 
of  ears  and  each  computes  an  average  length,  then,  in  a  sufficiently 
large  number  of  cases,  one  half  of  these  averages  will  be  within 
the  limits  set  and  one  half  will  be  without. 
In  treatises  on  probability  it  is  shown  that 

P          E,       0.6745o- 

\P)  &m  -  ~j=  =  — T=—  • 

Vn         Vn 

This  formula  shows  that  in  order  to  double  the  precision  of  the 
computed  arithmetic  average  it  is  necessary  to  take  four  times  as 
many  observations. 

231.  Probable  Error  in  the  Standard  Deviation.  Compute 
the  standard  deviation,  §  227,  of  the  lengths  of  500  ears  of  corn 
from  a  crib.  This  will  differ  slightly  from  the  true  standard 
deviation  <r,  of  the  lengths  of  all  the  ears  in  the  crib.  Next  find, 
by  means  of  equation  (6)  below,  the  probable  error  Eay  of  the 
standard  deviation.  Then  for  a  sufficiently  large  number  of 
samples  from  the  crib,  the  computed  standard  deviations  will 
fall  one  half  within  the  limits  a-  —  E*  and  a-  +  Ey,  and  one  half 
without.  The  formula  for  the  probable  error  in  the  standard 
deviation  is 
ttrt  F  .-  Em  _  0.6745Q- 

\P)  &v —  — j=—  . 

V2        V2n 

EXERCISES 

1.  Compute  E,,  Em,  E«  for  the  data  in  §  225. 

2.  Compute  <r,  C,  E,,  Em,  E<r  for  the  following  sets  of  measurements, 
(a)  5,  6,  7,  8,  8,  9,  10,  11 ;  (6)  5,  5,  5,  7,  9,  10,  11,  12. 

(c)   1,  6,  8,  8,  8,  8,  10,  15 ;  (d)  51,  56,  58,  58,  58,  58,  60,  65. 


302 


MATHEMATICS 


[XVIII,  §  231 


3.    Compute  <r,  C,  E,,  Em,  E*  for  the  following  distribution  of  oat 
plants  with  respect  to  height  in  centimeters  [LOVE-LEIGHTY]. 


Height  

60 

65 

70 

75 

80 

85 

90 

Frequency  .... 

2 

11 

45 

140 

122 

73 

7 

(a)- 


(6) 


Height 

60 

65 

70 

75 

80 

85 

90 

95 

Frequency.  .  .  . 

11 

36 

60 

94 

99 

102 

68 

18 

4.  Compute  from  the  following  data  the  mode,  the  mean,  the  coef- 
ficient of  variability,  the  standard  deviation,  the  probable  error  in  the 
mean,  and  the  probable  error  in  the  standard  deviation. 


Lbs.  of  butter  fat  .  . 
No.  of  cows  

400 
1 

375 

?, 

350 
4 

325 
5 

300 

7 

275 
6 

250 
5 

225 

2 

200 
1 

Draw  the  distribution  curve. 

5.  The  following  table  is  taken  from  BULLETIN  110,  PART  1,  Bureau 
of  Animal  Husbandry,  U.  S.  Dept.  of  Agriculture  on  "A  BIOMETRICAL 
STUDY  OF  EGG  PRODUCTION  IN  THE  DOMESTIC  FOWL"  and  shows  the 
frequency  distribution  for  hens  in  first-year  egg  production. 


Annual  Egg 
Production^ 

A 

£* 

IS 

M 

60 
7? 

If 

90 
ITJ? 

m 

120 
134 

m 

ill 

i-n 

180 

±y? 

195 
25'J 

fl  0 
23 

IIS 

1902-03 
1903-04 

7 

2 
5 

5 

1 

10 

5 
10 

8 
?0 

17 

?4 

18 
W 

17 

5? 

26 
37 

17 
W 

18 
16 

9 

8 

2 
2 

6 

1 

1905-06 

r 

?, 

4 

q 

13 

?5 

?4 

?,?, 

3?! 

17 

90 

q 

1906-07 
(a) 

2 

2 

5 

5 

q 

16 

30 

39 

?6 

?1 

1Q 

1? 

1 

00 

10 

8 

8 

15 

29 

32 

48 

39 

36 

25 

18 

6 

5 

2 

From  this  data  compute  for  each  year  the  mean,  the  median,  and  the 
mode  for  egg  production.  Compute  ff,  C,  E<r,  Em,  E,.  Draw  the  dis- 
tribution curve. 

6.  From  Table  I  at  the  end  of  Chapter  XIX  compute  for  each  weight 
(length)  the  mean,  the  median,  and  the  mode  for  length  (weight). 
Compute  <r,  C,  E<r,  Em,  E,  of  weight  (length)  for  each  length  (weight). 

7.  For  Table  II  (p.  312)  follow  the  directions  as  given  in  Ex.  6  for 
Table  I,  reading  however  number  of  kernels  instead  of  weight. 


XVIII,  §  231]  PROBABILITY  303 

8.  For  Table  III  (p.  312)  follow  the  directions  as  given  in  Ex.  6 
for  Table  I,  reading  yield  and  number  of  culms  in  place  of  weight 
and  length. 

9.  For  Table  IV  (p.  313)  follow  the  directions  as  given  in  Ex.  6. 
Read  height  of  mid-parent  and  height  of  adult  children  in  place  of 
weight  and  length. 


CHAPTER  XIX 
CORRELATION* 

232.  Meaning  of  Correlation.  Whenever  two  quantities 
are  so  related  that  an  increase  in  one  of  them  produces  or  is  ac- 
companied by  an  increase  in  the  other  and  the  greater  the  in- 
crease in  the  one  the  greater  the  increase  in  the  other,  these 
quantities  are  said  to  be  correlated  positively.  If  an  increase 
in  one  produces,  or  is  accompanied  by,  a  decrease  in  the  other, 
they  are  said  to  be  correlated  negatively.  If  a  change  in  one  is 
not  accompanied  by  any  change  in  the  other,  there  is  no  corre- 
lation, and  the  quantities  are  said  to  be  unrelated.  Perfect 
positive  correlation  is  represented  by  the  number  +  1,  perfect 
negative  correlation  by  —  1,  no  correlation  by  zero.  There 
is  perfect  positive  correlation  between  the  area  of  a  rectangular 
field  and  its  length,  the  extension  of  a  spiral  spring  and  the  sus- 
pended load.  There  is  perfect  negative  correlation  between 
the  pressure  and  volume  of  a  perfect  gas.  No  relation  exists 
between  the  price  of  coal  and  the  length  of  ears  of  corn. 

There  are  quantities,  common  in  everyday  life,  such  that  a 
change  in  one  is  not  accompanied  by  a  proportionate  change 
in  the  other,  but  a  given  change  in  one  is  always  accompanied  by 
some  change  in  the  other.  Such  quantities  are  still  said  to  be 
correlated.  The  degree  of  relationship  may  be  anywhere  be- 
tween complete  independence  and  complete  dependence,  that  is 

*  Throughout  this  Chapter,  the  authors  have  consulted  the  following  books,  and  are 
indebted  to  them  for  ideas:  E.  DAVENPORT,  Principles  of  Breeding  (Chap.  XIII); 
ZIZEK,  Statistical  Averages;  SECRIST,  Introduction  to  Statistical  Methods;  PEAKSON, 
Grammar  of  Science;  BOWLEV,  Elements  of  Statistics. 

304 


XIX,  §  232] 


CORRELATION 


305 


between  zero  and  +  1  or  between  zero  and  —  1.  For  example 
we  may  mention  the  effect  of  potato  prices  on  acreage,  and  vice 
versa. 

^Ve  desire  a  numerical  measure  for  this  correlation.  Any 
adequate  expression  must  be  such  that  it  becomes  zero  when 
there  is  no  correlation,  —  1  when  there  is  perfect  negative  corre- 
lation, +  1  for  perfect  positive  correlation,  and  which  is  always 
between  —  1  and  +  1.  Yule  has  proposed  a  formula  which 
satisfies  these  conditions.  Arrange  the  observed  data  with  refer- 
ence to  the  two  quantities  in  question  as  in  the  following  dia- 
gram : 


x  present. 

x  absent. 

y  present  

U 

V 

y  absent  

T 

S 

Then  a  measure  m  of  the  correlation  existing  is  given  by  the 
equation 

(1) 


If  either  r  or  v  is  zero 
If  either  u  or  s  is  zero 
If  us  =  rv 


us  +  rv 

m  =  +  1. 
m  =  —  1. 
m  =  0. 

EXERCISES 


1.   Compute  from  the  following  table  the  degree  of  effectiveness  of 
vaccination  against  diphtheria : 


Recoveries. 

Deaths. 

Vaccinated  . 

2843 

106 

Not  vaccinated 

254 

225 

2.   Compute  from  the  following  table  the  correlation  between  prohi- 
bition and  the  arrests  per  day  in  a  given  city  for  one  year : 


306 


MATHEMATICS 


[XIX,  §  232 


Days  with  more  than  20  arrests. 

Less  than  20. 

Wet  

281 

84 

Dry 

142 

223 

3.  Compute  the  correlation  between  use  of  fertilizer  and  yield  of 
potatoes  in  bushels  per  acre  when  the  results  from  fifty  plats  are  as 
follows : 


Yield  over  100  bushels.) 

Under  100  bushels. 

Fertilizer. 

47 

3 

No  fertilizer  

14 

36 

Ans.   0.95 

This  high  value  of  correlation  is  considered  evidence  of  some  connec- 
tion between  use  of  fertilizer  and  yield. 

233.  Correlation  Table.  Let  it  be  proposed  to  find  the  de- 
gree of  correlation,  if  any,  between  the  lengths  of  ears  of  corn  and 
their  weight,  between  their  lengths  and  number  of  rows  of 
kernels,  between  length  and  circumference,  between  length  and 
yield  per  acre,  between  length  of  head  of  wheat  and  yield  per 
acre,  between  height  of  wheat  and  yield  per  acre.  The  problem 
is  now  more  complex.  Let  us  take  for  example  a  given  number 
of  ears  of  corn  and  examine  them  as  to  weight  in  ounces  and 
length  in  inches.  The  measurements  may  be  tabulated  as  shown 
in  the  accompanying  table.  Each  column  is  a  frequency  dis- 
tribution of  lengths  for  a  constant  weight.  Each  row  is  a  fre- 
quency distribution  of  weights  for  a  constant  length.  The 
distribution  of  the  ears  of  length  8  inches  with  respect  to  weight 
is  3,  7,  19,  25,  17,  22,  17,  3,  1. 

It  is  to  be  noticed  that  the  table  extends  across  the  enclosing 
rectangle  from  the  upper  left-hand  corner  to  the  lower  right- 
hand  corner.  Whenever  data  tabulated  with  respect  to  two 
measurable  characters  show  this  skew  arrangement,  correlation 
exists.  In  the  accompanying  table  weights  increase  from  left 


XIX,  §234] 


CORRELATION 


307 


to  right  and  lengths  increase  as  we  move  downward.  We 
have  in  this  case  positive  correlation.  An  extension  of  the  array 
from  the  upper  right-hand  corner  to  the  lower  left  would  have 
indicated  negative  correlation. 

234.   Coefficient  of  Correlation.      The  method  of  obtain- 
ing the  correlation  coefficient  may  be  explained  in  connection 

CORRELATION  BETWEEN  WEIGHT  AND  LENGTH  OF  EAR  * 


Weight  of  Ear  in  Ounces. 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

is 

19 

20 

21 

3 

1 

2 

1 

3.5 

4 

1 

4 

3 

5 

5 

1 

o5 

4.5 

6 

5 

4 

1 

QJ 

£ 

5 

2 

4 

7 

2 

4 

o 
a 

5.5 

2 

9 

15 

14 

8 

4 

1 

s 

6 

1 

2 

12 

16 

13 

13 

6 

1 

a 

«iH 

6.5 

1 

6 

11 

26 

11 

8 

6 

1 

03 

7 

1 

2 

2 

12 

18 

12 

12 

11 

4 

1 

H 

7.5 

1 

2 

4 

20 

12 

13 

21 

11 

6 

6 

1 

1 

•s 

8 

3 

7 

19 

25 

17 

22 

17 

3 

1 

,D 

8.5 

1 

1 

12 

9 

23 

30 

26 

26 

5 

1 

1 

9 

1 

7 

10 

23 

35 

26 

24 

12 

1 

2 

1 

a 

CJ 

9.5 

1 

4 

14 

19 

29 

17 

10 

1 

3 

1 

1 

^ 

10 

1 

1 

3 

8 

18 

10 

(I 

4 

2 

10.5 

2 

3 

6 

7 

2 

5 

1 

11 

1 

1 

2 

1 

11.5 

] 

with  the  above  table.  Find  the  arithmetic  mean  of  each  char- 
acter involved  —  in  this  case  mean  length  of  ear,  MI,  and  mean 
weight  of  ears,  Mw.  Find  the  deviation  DI  of  ear  length  from 
mean  length,  and  the  deviation  Dw  of  weight  from  mean  weight, 
for  each  ear  tabulated.  For  each  ear  tabulated  find  the  product 
of  DI  and  Dw  and  then  add  all  of  these  products.  This  sum  we 
will  indicate  by  3DiDw.  Find  in  the  usual  way  the  standard 
deviation  of  length  of  ears,  <TI,  and  the  standard  deviation  of 
weight  of  ears,  <rw.  Then  the  coefficient  of  correlation,  r,  is 

*  E.  DAVENPORT,  Principles  of  Breeding,  p.  458. 


308 


MATHEMATICS 


[XIX,  §  234 


given  by  the  formula 

(2) 


where  n  is  the  number  of  things  observed,  in  this  case  the  total 
number  of  ears. 

A  convenient  arrangement  for  computing  DI  for  each  ear  length  and 
Dw  for  each  ear  weight  is  shown  in  the  table  below. 

The  row  labeled  6.5  inches  (table  §  233),  gives  the  frequency  distri- 
bution of  ears  with  respect  to  weight.      There  is  one  ear  of  weight 
4  oz.,  6  ears  of  weight  5  oz.,  11  ears  of  weight  6  oz.,  26  ears  of  weight 
7  oz.,  etc.;  a  total  of  70  ears,  fi,  of  length  6.5  inches. 
fiVi  =  1X4  +  6X5  +  11X6  +  26  X7  +  11X8  +  8X9 

+  6  X  10  +  1  X  11  =  455.0 

The  mean  length  of  ear  is  obtained  by  adding  the  numbers  in  the 
column  headed  fiVi  and  dividing  this  sum  by  the  total  number, 
n  =  993,  of  ears. 

CORRELATION  OF  WEIGHT  TO  LENGTH  OF  EARS  OF  CORN 


Length, 

ji 

SiVi 

DI 

Weight, 

t 

fvVw 

Dw 

DiDv 

Inches. 

Ounces. 

3 

4 

12.0 

-  4.8 

2 

4 

8 

-8.7 

143.0 

3.5 

5 

17.5 

-  4.3 

3 

22 

66 

-7.7 

156.9 

4 

14 

56.0 

-  3.8 

4 

27 

108 

-6.7 

394.4 

4.5 

16 

72.0 

-  3.3 

5 

50 

250 

-5.7 

347.2 

5 

19 

95.0 

-  2.8 

6 

47 

282 

-4.7 

297.6 

5.5 

53 

291.5 

-  2.3 

7 

71 

497 

-3.7 

618.9 

6 

64 

384.0 

-  1.8 

8 

75 

600 

-2.7 

465.8 

6.5 

70 

455.0 

-  1.3 

9 

71 

639 

-1.7 

306.8 

7 

75 

525.0 

-  0.8 

10 

75 

750 

-0.7 

110.8 

7.5 

98 

735.0 

-  0.3 

11 

88 

968 

0.3 

14.9 

8 

114 

912.0 

0.2 

12 

107 

1,284 

1.3 

1.4 

8.5 

134 

1,139.0 

0.7 

13 

114 

1,482 

2.3 

129.6 

9 

142 

1,278.0 

1.2 

14 

112 

1,568 

3.3 

466.3. 

9.5 

100 

950.0 

1.7 

15 

65 

975 

4.3 

564.4 

10 

53 

530.0 

2.2 

16 

37 

592 

5.3 

431.0 

10.5 

26 

273.0 

2.7 

17 

8 

136 

6.3 

364.0 

11 

5 

55.0 

3.2 

18 

13 

234 

7.3 

107.2 

11.5 

1 

11.5 

3.7 

19 

4 

76 

8.3 

27.0 

993 

7,791.5 

20 

2 

40 

9.3 

21 

1 

21 

10.3 

M 

7791.5 

5 

993 

10,576 

1         993         ™ 

Mw 

10,576 

=  10.65 

993 

*  E.  DAVENPORT,  Principles  of  Breeding,  p.  461. 


XIX,  §  235]  CORRELATION  309 

All  of  the  symbols  used  have  been  defined  with  the  exception  of  the 
following :  <r/  is  the  standard  deviation  of  length ;  /„,  is  the  number  (fre- 
quency) of  ears  of  same  weight  w ;  Vi  stands  for  the  value  of  length  of 
ears  with  given  frequency ;  Vw  represents  the  value  of  weight  of  ears 
with  given  frequency.  This  gives  MI  =  7.85.  In  the  row  labeled  6.5 
and  in  the  column  headed  DI  we  write  the  difference  between  this  mean 
length  7.85  and  the  length  6.5.  This  gives  the  number  —  1.3  of  the  col- 
umn headed  D/.  The  number  306.8  in  the  last  column  is  obtained  as 
follows : 

(-  1.3)[1(-  6.7)  +  6(-  5.7)  +  11(-  4.7)  +  26(-  3.7) 

+  11(-  2.7)  +  8(-  1.7)  +  6(-  0.7)  +  1(0.3)]  =  306.8 

That  is,  the  ear  of  weight  4  oz.  deviates  from  the  mean  weight  by  6.7  oz., 
the  6  ears  of  weight  5  oz.  deviate  from  the  mean  weight  by  5.7  oz., 
the  11  ears  of  weight  6  oz.  deviate  from  the  mean  weight  by  4.7  oz., 
etc. 

The  number  306.8  represents  the  sum  of  the  products  of  the  cor- 
responding length  and  weight  deviations  for  every  individual  in  the 
horizontal  row  to  which  the  number  belongs.  To  find  the  correlation 
coefficient  add  the  numbers  in  the  column  headed  DiDw,  obtaining  in 
this  case  4947.2. 

Divide  this  number  4947.2  by  n  X  <TI  X  «•„.  In  this  case  n  =  993, 
and  <TI,  ffw  have  been  computed  to  be  1.57  and  3.63  respectively. 
This  gives  the  correlation  coefficient 

-  -         4947'2          =  0.87 


993(1.57)  (3.63) 

235.  The  Regression  Curve.  For  each  recorded  height  (see 
table,  §  233)  compute  the  arithmetic  average  of  length  of  ears. 
Thus  the  ears  of  weight  4  oz.  have  an  average  length  of  5.1  inches. 
The  ears  of  weight  5  oz.  have  an  average  length  of  5.46  inches, 
etc.  Plot  a  curve  using  for  abscissas  the  weights,  and  for 
ordinates  the  computed  average  lengths.  The  curve  so  plotted 
is  called  a  regression  curve.  In  many  cases  this  curve  is  a 
straight  line.  It  can  be  shown  that  the  straight  line  which  best 
represents  the  plotted  data  is  given  by  the  equation 


310  MATHEMATICS  [XIX,  §  235 

(2)  Mt  =  r^lw. 

aw 

Another  regression  curve  can  be  plotted  for  the  same  data, 
using  lengths  as  ordinates  and  mean  weights  for  abscissas.  This 
curve  does  not  in  general  coincide  with  the  first.  Its  equation  is 

Mw  =  r^l. 
&1 

By  means  of  these  curves  the  mean  value  of  one  character  can  be 
read  off  when  a  fixed  value  is  given  to  the  other  character. 

EXERCISES 

1 .  Find,  for  the  correlation  table  in  §  233  : 

(a)  the  regression  of  weight  relative  to  length ; 

(b)  regression  of  length  relative  to  weight. 

Ans.    (a)  2.03     (6)  0.38 

2.  Find  the  equation  of  the  line  of  regression  in  both  cases  of  Ex.  1. 

3.  Plot  the  line  of  regression  in  Ex.  2  from  the  equation  found  there 
and  then  again  plot  the  line  from  the  data  as  suggested  in  §  235. 

4.  From  Table  II,  p.  312,  which  gives  the  correlation  of  height  of  oat 
plants  with  the  average  number  per  plant  of  kernels  per  culm,  compute 
the  mean  height,  the  mean  number  of  kernels  per  culm,  the  standard 
deviation  with  respect  to  height,  the  standard  deviation  with  respect  to 
number  of  kernels  per  culm,  the  correlation  coefficient,  and  the  regres- 
sion coefficients. 

5.  Examine  Table  IV,  p.  313,  which  gives  the  number  of  children  of 
various  statures  born  of  205  mid-parents  of  various  statures.     From 
this  table  compute : 

Mp  =  mean  height  of  mid-parents, 

Me  —  mean  height  of  adult  children, 
ffp  =  standard  deviation  of  height  of  mid-parents, 
ffe  =  standard  deviation  of  height  of  adult  children, 
r  =  the  correlation  coefficient,  and  both  regression  coefficients. 

6.  For  Ex.  4  plot  the  lines  of  regression  (a)  from  their  equations, 
(6)  from  the  data  directly. 

7.  For  Ex.  5  plot  the  lines  of  regression  (a)  from  their  equations, 
(6)  from  the  data  directly. 


XIX,  §235] 


CORRELATION 


311 


8.  From  the  following  table  find  a  measure  of  the  effectiveness  of 
vaccination  against  smallpox. 


Recoveries. 

Deaths. 

Total. 

Vaccinated  

3,951 

200 

4,151 

Not  vaccinated  

278 

274 

552 

Total  

4,229 

474 

4,703 

9.  Construct  a  correlation  table  from  your  own  observations  on 
length  and  breadth  of  leaves,     (a)  Use  30  classes  for  length,     (fe)  Use 
15  classes  for  length,  thus  making  the  class  interval  twice  as  large. 
Compute  in  each  case  the  correlation  coefficient. 

10.  From  Table  I,  below,  which  gives  the  conslation  of  lengths  and 
weights  of  ears  of  corn,  compute  the  mean  length,  the  mean  weight, 
the  standard  deviation  with  respect  to  length,  the  standard  deviation 
with  respect  to  weight,  the  correlation  coefficient,  and  both  regression 
coefficients. 

11.  The  same  as  Ex.  10  after  writing  number  of  kernels  in  place  of 
weight,  using  Table  II,  p.  312,  in  place  of  Table  I. 

I.     CORRELATION  OP  LENGTH  AND  WEIGHT  OF  EARS  OF  CORN 


Length 
In  Inches. 

Weight  In  Ounces. 

2 

3 

4 

5 

c 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

is 

3.0 

1 

1 

1 

3.5 

1 

2 

2 

1 

4.0 

2 

3 

5 

4 

1 

4.5 

4 

5 

6 

2 

1 

5.0 

4 

7 

8 

6 

4 

1 

5.5 

3 

9 

12 

13 

8 

3 

1 

6.0 

1 

5 

10 

15 

12 

9 

5 

2 

6.5 

2 

6 

12 

26 

14 

10 

5 

3 

1 

7.0 

1 

3 

4 

14 

18 

15 

10 

7 

2 

1 

7.5 

1 

2 

6 

13 

17 

19 

13 

9 

6 

4 

2 

8.0 

2 

7 

10 

13 

19 

7 

6 

2 

1 

8.5 

1 

3 

9 

14 

25 

17 

8 

5 

1 

9.0 

1 

4 

7 

19 

25 

16 

11 

3 

9.5 

2 

3 

8 

18 

20 

15 

6 

1 

10.0 

1 

3 

9 

18 

13 

7 

5 

•2 

10.5 

2 

3 

7 

5 

4 

1 

11.0 

1 

2 

3 

2 

312 


MATHEMATICS 


[XIX,  §  235 


II.  CORRELATION  OP  AVERAGE  HEIGHT  OF  OAT  PLANTS  IN  CENTI- 
METERS AND  AVERAGE  NUMBER  OF  KERNELS  PER  CULM  PER 
PLANT.  [LOVE-LEIGHTY.]  r  =  0.73. 


Number  of  Kernels. 

Height. 

30 

40; 

50 

GO 

70 

80 

90 

100 

110 

120 

40 

50 

60 

70 

80 

90 

100 

110 

120 

130 

55-60 

1 

1 

60-65 

4 

7 

65-70 

7 

22 

9 

6 

1 

70-75 

1 

13 

30 

59 

32 

5 

75-80 

2 

16 

40 

38 

23 

3 

80-85 

1 

12 

26 

23 

9 

2 

85-90 

3 

2 

2 

III.  CORRELATION  OF  NUMBER  OF  CULMS  PER  OAT  PLANT  AND 
TOTAL  YIELD  OF  PLANT  IN  GRAMS.  [LOVE-LEIGHTY.] 
r  =  0.712 


Yield 

Number  of  Culms  per  Plant. 

2 

3 

4 

5 

6 

7 

0-1  

3 

28 
18 
1 

19 
66 

42 

7 

3 
20 
58 
59 
26 

1 

1 

7 
11 
14 
4 

1 

1 

3 
2 
3 

1 

1 

1-2  

2-3    

3-4  

4-5    . 

5-6  

6-7  

7-8   . 

8-9  

XIX,  §235] 


CORRELATION 


313 


IV.     CORRELATION  OF  HEIGHTS  OF  ADULT  CHILDREN  AND  PARENTS 
DATA  FOR  CHILDREN  OF  205  MID-PARENTS*  OF  VARIOUS  STATURES 


Heights  of 
Mid-parents. 

Heights  of  Adult  Children  In  Inches. 

Above. 

73.2 

72.2 

71.2 

70.2 

69.2 

68.2 

67.2 

66.2 

65.2 

64.2 

63.2 

62.2 

Below. 

Above 

3 

1 

72.5 

4 

2 

7 

2 

1 

2 

1 

71.5 

2 

2 

9 

4 

10 

5 

3 

4 

3 

1 

70.5 

3 

3 

4 

7 

14 

18 

12 

3 

1 

1 

1 

1 

69.5 

5 

4 

11 

20 

25 

33 

20 

27 

17 

4 

16 

1 

68.5 

3 

4 

18 

21 

48 

34 

31 

25 

16 

11 

7 

1 

67.5 

4 

11 

19 

38 

28 

38 

36 

15 

14 

5 

3 

66.5 

4 

13 

14 

17 

17 

2 

5 

3 

3 

65.5 

1 

2 

5 

7 

7 

11 

11 

7 

5 

9 

1 

64.5 

5 

5 

1 

4 

4 

1 

1 

Below 

1 

1 

2 

2 

1 

4 

2 

1 

*  Height  of  mid-parent  is  the  mean  height  of  the  two  parents. 
[GALTON-DAVENPORT] 


GREEK  ALPHABET 


LETTERS  NAMES 


A  a     Alpha 

H, 

B  0     Beta 

90 

T  7     Gamma 

It 

A  6     Delta 

KK 

E  €     Epsilon 

AX 

Z  f     Zeta 

MM 

LETTERS  NAMES 

LETTERS  NAMES 

N  v 

Nu 

TT     Tau 

H£ 

Xi 

T  v      Upsilon 

Oo 

Omicron 

€>  0     Phi 

UTT 

Pi 

XX     Chi 

Pp 

Rho 

¥  \ff    Psi 

So-s 

Sigma 

Q  w     Omega 

314 


FOUR  PLACE  TABLES 

PAGES 

I.  LOGARITHMS  OF  NUMBERS 316-319 

II.  VALUES  AND    LOGARITHMS  OF  TRIGONOMETRIC 

FUNCTIONS 320-324 

III.  RADIAN  MEASURE — TRIGONOMETRIC  FUNCTIONS          325 

IV.  SQUARES,  CUBES,  SQUARE  ROOTS,  CUBE  ROOTS  326 
V.  IMPORTANT  CONSTANTS 327 

VI.  DEGREES  TO  RADIANS 327 

VII.  COMPOUND  INTEREST 328 

VIII.  AMERICAN  EXPERIENCE  MORTALITY  TABLE.  .  .  32!) 

IX.  HEIGHTS  AND  WEIGHTS  OF  MEN 330 

EXPLANATION  OF  TABLE  II .                                     .  331-333 


315 


Table  I.     Logarithms  of  Numbers 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8    9 

Prop.  Parts 

0 

0000 

3010 

4771 

6021 

6990 

7782 

8451 

9031 

9542 

22 

21 

1 

0000 

0414 

0792 

1139 

1461 

1761 

2041 

2304 

2553 

2788 

l 

2.2 

2.1 

2 

3010 

3222 

3424 

3617 

3802 

3979 

4150 

4314 

4472 

4624 

2 

4.4 

4.2 

3 

4771" 

4914 

5051 

5185 

5315 

5441 

5563 

5682 

5798 

5911 

3 

4 

6.6 
8.8 

6.3 
8.4 

5 

11.0 

10.5 

4 

6021 

6128 

6232 

6335 

6435 

6532 

6628 

6721 

6812 

6902 

6 

13.2 

12.6 

5 

6990 

7076 

7160 

7243 

7324 

7404 

7482 

7559 

7634 

7709 

7 

15.4 

14.7 

6 

7782 

7853 

7924 

7993 

8062 

8129 

8195 

8261 

8325 

8388 

8 
9 

17.6 
19.8 

16.8 
18.9 

7 

8451 

8513 

8573 

8633 

8692 

8751 

8808 

8865 

8921 

8976 

20 

19 

8 

9031 

9085 

9138 

9191 

9243 

9294 

9345 

9395 

9445 

9494 

1 

2.0 

1.9 

9 

9542 

9590 

9638 

9685 

9731 

9777 

9823 

9868 

9912 

9956 

2 
3 

4.0 
6  0 

3.8 

5  7 

10 

0000 

0043 

"0086 

0128 

0170" 

0212 

0253" 

0294 

0334 

0374 

4 

8.0 

7^6 

11 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

5 

10.0 

9.5 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

6 

7 

12.0 
14  0 

11.4 
13  3 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

8 

ie!o 

15.2 

9  118.0 

17.1 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

IB 

17 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

1 

lo 

1.8 

i  / 

1  7 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

2 

3.6 

3.4 

3 

5.4 

5.1 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

4 

7.2 

6.8 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

5 

9.0 

8.5 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

6 

7 

10.8 
12.6 

10.2 
11.9 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

8 

14^4 

13.6 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

9 

16.2 

15.3 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

16 

15 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

1 

1.6 

1.5 

2 

3.2 

3.0 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

3 

4.8 

4.5 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

4 
5 

6.4 
8.0 

6.0 
7.5 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

6 

9^6 

9^0 

7 

11.2 

10.5 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

8 

12.8  12.0 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

9 

14.4  13.5 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

14 

13 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

1 

1.4 

O  o 

1.3 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

3 

—  .O 

4.2 

2.6 
3.9 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

4 

5.6 

5.2 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

5 

7.0 

6.5 

6 

8.4 

78 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

7 

9.8 

9.1 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

8 
9 

11.2 
12.6 

10.4 
11.7 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

12 

11 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

1 

1  2 

1.1 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

2 
3 

2.4 
3.6 

2.2 
3.3 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

4 

4^8 

4.4 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

5 

6.0 

7  A 

5.5 

ft  ft 

41 

6128 

6138 

6149 

6160  ;  6170 

6180 

6191 

6201 

6212 

6222 

7 

.« 

8.4 

D.n 

7.7 

42 

6232 

6243 

6253 

6263  6274 

6284 

6294 

6304 

6314 

6325 

8 

9.6 

8.8 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

9 

10.8 

9.9 

9 

8 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

1 

0.9 

0.8 

45 

6532 

6542 

6551 

6561  6571 

6580 

6590 

6599 

6609 

6618 

2 

1.8 

1.6 

46 

6628 

6637 

6646 

6656  6665 

6675 

6684 

6693 

6702 

6712 

3 

2.7 

2.4 

4 

3.6 

3.2 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776  6785 

6794 

6803 

5 

4.5 

4.0 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866  6875 

6884 

6893 

6 

7 

5.4 
6.3 

4.8 
5.6 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955  6964 

6972 

6981 

8 

7^2 

6^4 

50 

6990 

6998 

7007 

"7016" 

7024 

7033 

7042  7050 

7059 

7067 

9   8.1 

7.2 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

316 


Table  I.    Logarithms  of  Numbers 


N. 

0 

1 

2 

3 

4 

5 

6  |  7 

8 

9 

Prop.  Parts 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

9 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

2 

1  8 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

3 

2.7 

4 

3.6 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

5 

4.5 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

6 

5.4 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

7 
8 

6.3 

7.2 

9 

8.1 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760  7767 

7774 

8 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

1 

0.8 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

3 

2.4 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

4 

3.2 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

5 

4.0 

6 

4.8 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

7 

5.0 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

8 
g 

7  9 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

7 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363  8370  8376 

8382 

1 

0.7 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426  8432  8439 

8445 

2 

1.4 

70 

8451 

8457 

8463 

8470 

8476 

84  S2 

8488  8494 

8500 

8506 

3 
4 

2.1 
2.8 

71 

8513 

8519 

8525 

8531 

8537 

85-13 

8549 

8555 

8561 

8567 

5 

3.5 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

6 

4.2 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

7 

4.9 

8 

5.6 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

9 

6.3 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

6 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

1 
2 

0.6 
1.2 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

3 

1.8 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

4 

2.4 

80 

!K.)31 

9036  9042 

9047 

9053 

905X 

9063 

9069 

9074 

9079 

6 

3.6 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

7 

4.2 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

8 

4.8 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

9 

5.4 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

2 

1.0 

3 

1  5 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

4 

2.0 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479  ,  9484 

9489 

5 

2.5 

89 

9494 

9499 

'.).-,(  )4 

9509 

9513 

9518 

9523 

9528 

9533 

053S 

e 

3.0 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

95S1 

9586 

8 

4.0 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

!)<>•_>  i 

9<>L>S 

9633 

9 

4.5 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

4 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

1 

04 

95 

9777 

9782 

9786 

9791  |  9795 

9800 

9805 

9809  9814  9818 

3 

1  2 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

4 

1.6 

5 

2.0 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

G 

2.4 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

7 

2.8 

99 

9956 

9961 

<)'.)<  if, 

9969 

9974 

0!>7S 

9983  ;  9987  ;  9991 

9996 

8 
9 

3.2 
3.6 

100 

0000 

0004 

0009 

0013 

0017 

0022 

0026  ;  0030 

0035 

0039 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

317 


Table  I.     Logarithms  of  Numbers 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Parts 

100 

0000 

0004 

0009 

0013 

0017 

0022 

0026 

0030 

0035 

0039 

101 

0043 

0048 

0052 

0056 

0060 

0065 

0069 

0073 

0077 

0082 

102 

0086 

0090 

0095 

0099 

0103 

0107 

0111 

0116  ,0120 

0124 

103 

0128 

0133 

0137 

0141 

0145 

0149 

0154 

0158 

0162 

0166 

104 

0170 

0175 

0179 

0183 

0187 

0191 

0195 

0199 

0204 

0208 

105 

0212 

0216 

0220 

0224 

0228 

0233 

0237 

0241 

0245 

0249 

106 

0253 

0257 

0261 

0265 

0269 

0273 

0278 

0282 

0286 

0290 

5 

1 

0.5 

107 

0294 

0298 

0302 

0306 

0310 

0314 

0318 

0322 

0326 

0330 

2 

1.0 

108 

0334 

0338 

0342 

0346 

0350 

0354 

0358 

0362 

0366 

0370 

3 

1.5 

109 

0374 

0378 

0382 

0386 

0390 

0394 

0398 

0402 

0406 

0410 

5 

2.5 

110 

0414 

0418 

0422 

0426 

0430 

0434 

0438 

0441 

0445 

0449 

6 

3.0 

111 

0453 

0457 

"0461 

04~65" 

0469" 

0473" 

0477 

0481 

0484 

0488 

7 
g 

3.5 
•1  0 

112 

0492 

0496 

0500 

0504 

0508 

0512 

0515 

0519 

0523 

0527 

9 

4.5 

113 

0531 

0535 

0538 

0542 

0546 

0550 

0554 

0558 

0561 

0565 

114 

0569 

0573 

0577 

0580 

0584 

0588 

0592 

0596 

0599 

0603 

115 

0607 

0611 

0615 

0618 

0622 

0626 

0630 

0633 

0637 

0641 

116 

0645 

0648 

0652 

0656 

0660 

0663 

0667 

0671 

0674 

0678 

4 

117 

0682 

0686 

0689 

0693 

0697 

0700 

0704 

0708 

0711 

0715 

1 

0.4 

118 

0719 

0722 

0726 

0730 

0734 

0737 

0741 

0745 

0748 

0752 

2 

0.8 

119 

0755 

0759 

0763 

0766 

0770 

0774 

0777 

0781 

0785 

0788 

3 

1.2 

120 

0792 

0795 

0799 

0803 

0806 

0810 

0813 

0817 

0821 

0824 

5 

2.0 

121 

0828 

0831 

0835 

0839 

0842 

0846 

0849 

0853 

0856 

0860 

6 

2.4 

122 

0864 

0867 

0871 

0874 

0878 

0881 

0885 

0888 

0892 

0896 

7 

2.8 

123 

0899 

0903 

0906 

0910 

0913 

0917 

0920 

0924 

0927 

0931 

8 
9 

3.2 
3.6 

124 

0934 

0938 

0941 

0945 

0948 

0952 

0955 

0959 

0962 

0966 

125 

0969 

0973 

0976 

0980 

0983 

0986 

0990 

0993 

0997 

1000 

126 

1004 

1007 

1011 

1014 

1017 

1021 

1024 

1028 

1031 

1035 

127 

1038 

1041 

1045 

1048 

1052 

1055 

1059 

1062 

1065 

1069 

128 

1072 

1075 

1079 

1082 

1086 

1089 

1092 

1096 

1099 

1103 

3 

129 

1106 

1109 

1113 

1116 

1119 

1123 

1126 

1129 

1133 

1136 

2 

0.6 

130 

1139 

1143 

1146 

1149 

1153 

1156 

1159 

1163 

1166 

1169 

3 

0.9 

131 

1173 

1176 

1179 

1183 

1186 

1189 

1193 

1196 

1199 

1202 

5 

1  5 

132 

1206 

1209 

1212 

1216 

1219 

1222 

1225 

1229 

1232 

1235 

6 

1.8 

133 

1239 

1242 

1245 

1248 

1252 

1255 

1258 

1261 

1265 

1268 

7 

2.1 

8 

2.4 

134 

1271 

1274 

1278 

1281 

1284 

1287 

1290 

1294 

1297 

1300 

9 

2.7 

135 

1303 

1307 

1310 

1313 

1316 

1319 

1323 

1326 

1329 

1332 

136 

1335 

1339 

1342 

1345 

1348 

1351 

1355 

1358 

1361 

1364 

137 

1367 

1370 

1374 

1377 

1380 

1383 

1386 

1389 

1392 

1396 

138 

1399 

1402 

1405 

1408 

1411 

1414 

1418 

1421 

1424 

1427 

139 

1430 

1433 

1436 

1440 

1443 

1446 

1449 

1452 

1455 

1458 

2 

140 

1*461 

1464 

1467 

1471 

1474 

1477 

1480 

1483 

1486 

1489 

o 

0.4 

141 

1492 

1495 

1498 

1501 

1504 

1508 

1511 

1514 

1517 

1520 

3 

0.6 

142 

1523 

1526 

1529 

1532 

1535 

1538 

1541 

1544 

1547 

1550 

4 

0.8 

143 

1553 

1556 

1559 

1562 

1565 

1569 

1572 

1575 

1578 

1581 

5 
6 

1.0 
1.2 

7 

1  4 

144 

1584 

1587 

1590 

1593 

1596 

1599 

1602 

1605 

1608 

1611 

8 

1.6 

145 

1614 

1617 

1620 

1623 

1626 

1629 

1632 

1635 

1638 

1641 

9 

l.S 

146 

1644 

1647 

1649 

1652 

1655 

1658 

1661 

1664 

1667 

1670 

147 

1673 

1676 

1679 

1682 

1685 

1688 

1691 

1694 

1697 

1700 

148 

1703 

1706 

1708 

1711 

1714 

1717 

1720 

1723 

1726 

1729 

149 

1732 

1735 

1738 

1741 

1744 

1746 

1749 

1752 

1755 

1758 

150 

1761 

1764 

1767 

1770 

1772 

1775 

1778 

1781 

1784 

1787 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

318 


Table  I.     Logarithms  of  Numbers 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Parts 

150 

1761 

1764 

1767 

1770 

1772 

1775 

1778 

1781 

1784 

1787 

151 

1790 

1793 

1796 

1798 

1801 

1804 

1807 

1810 

1813 

1816 

152 

1818 

1821 

1824 

1827 

1830 

1833 

1836 

1838 

1841 

1844 

153 

1847 

1850 

1853 

1855 

1858 

1861 

1864 

1867 

1870 

1872 

154 

1875 

1878 

1881 

1884 

1886 

1889 

1892 

1895 

1898 

1901 

155 

1903 

1906 

1909 

1912 

1915 

1917 

1920 

1923 

1926 

1928 

156 

1931 

1934 

1937 

1940 

1942 

1945 

1948 

1951 

1953 

1956 

3 

1     0.3 

157 

1959 

1962 

1965 

1967 

1970 

1973 

1976 

1978 

1981 

1984 

2     0.6 

158 

1987 

1989 

1992 

1995 

1998 

2000 

2003 

2006 

2009 

2011 

3     0.9 

159 

2014 

2017 

2019 

2022 

2025 

2028 

2030 

2033 

2036 

2038 

4     1.2 
5     15 

160 

2041 

2044 

2047 

2049 

2052 

2055 

2057 

2060 

2063 

20(10 

6     1.8 

161 

2068 

2071 

2074 

2076 

2079 

2082 

2084 

2087 

2090 

2092 

7     2.1 

162 

2095 

2098 

2101 

2103 

2106 

2109 

2111 

2114 

2117 

2119 

9     2.7 

163 

2122 

2125 

2127 

2130 

2133 

2135 

2138 

2140 

2143 

2146 

164 

2148 

2151 

2154 

2156 

2159 

2162 

2164 

2167 

2170 

2172 

165 

2175 

2177 

2180 

2183 

2185 

2188 

2191 

2193 

2196 

2198 

166 

2201 

2204 

2206 

2209 

2212 

2214 

2217 

2219 

2222 

2225 

167 

2227 

2230 

2232 

2235 

2238 

2240 

2243 

2245 

2248 

2251 

168 

2253 

2256 

2258 

2261 

2263 

2266 

2269 

2271 

2274 

2276 

169 

2279 

2281 

2284 

2287 

2289 

2292 

2294 

2297 

2299 

2302 

170 

2304 

2307 

2310  2312 

2315 

2317 

2320 

2322 

2:525 

2327 

171 

2330 

2333 

2335 

2338 

2340 

2343 

2345 

2348 

2350 

2353 

172 

2355 

2358 

2360 

2363 

2365 

2368 

2370 

2373 

2375 

2378 

173 

2380 

2383 

2385 

2388 

2390 

2393 

2395 

2398 

2400 

2403 

174 

2405 

2408 

2410 

2413 

2415 

2418 

2420 

2423 

2425 

2428 

175 

2430 

2433 

2435 

2438 

2440 

2443 

2445 

2448 

2450 

2453 

176 

2455 

2458 

2460 

2463 

2465 

2467 

2470 

2472 

2475 

2477 

2 

1     0.2 

177 

2480 

2482 

2485 

2487 

2490 

2492 

2494 

2497 

2499 

2502 

2     0.4 

178 

2504 

2507 

2509 

2512 

2514 

2516 

2519 

2521 

2524 

2526 

3     0.6 

179 

2529 

2531 

2533 

2.-,:;(> 

2538 

2541 

2543 

2545 

2548 

2550 

5     1.0 

180 

2553 

2555 

2558 

2560 

2562 

2565 

2567 

2570 

2572 

2574 

6     1.2 

181 

2577 

2579 

2582 

2584 

2586 

2589 

2591 

2594 

2596 

2598 

7     1.4 
8     16 

182 

2601 

2603 

2605 

2608 

2610 

2613 

2615 

2617 

2620 

2622 

9     1.8 

183 

2625 

2627 

2629 

2632 

2634 

2636 

2639 

2641 

2643 

2646 

184 

2648 

2651 

2653 

2655 

2658 

2660 

2662 

2665 

2667 

2669 

185 

2672 

2674 

2676 

2679 

2681 

2683 

2686 

2688 

2690 

2693 

186 

2695 

2697 

2700 

2702 

2704 

2707 

2709 

2711 

2714 

2716 

187 

2718 

2721 

2723 

2725 

2728 

2730 

2732 

2735 

2737 

2739 

188 

2742 

2744 

2746 

2749 

2751 

2753 

2755 

2758 

2760 

2762 

189 

2765 

2767 

2769 

2772 

2774 

2776 

2778 

2781 

2783 

2785 

190 

27SH 

2790 

2792 

2794 

2797 

2799 

2801 

2804 

2806 

2808 

191 

2810 

2813 

2815 

2817 

2819 

2822 

2824 

2S2(> 

2S2S 

2831 

192 

2833 

2835 

2838 

2840 

2842 

2844 

2847 

2849 

2851 

2853 

193 

2856 

2858 

2860 

2862 

2865 

2867 

2869 

2871 

2874 

2876 

194 

2878 

2880 

2882 

2885 

2887 

2889 

2891 

2894 

2896 

2898 

195 

2900 

2903 

2905 

2907 

2909 

2911 

2914 

2916 

2918 

2920 

196 

2923 

2925 

2927 

2929 

2931 

2934 

2936 

2938 

2940 

2942 

197 

2945 

2947 

2949 

2951 

2953 

2956 

2958 

2960 

2962 

2964 

198 

2967 

2969 

2971 

2973 

2975 

2978 

2980 

2982 

2984 

2986 

199 

2989 

2991 

2993 

2995 

2997 

21MHI 

3002 

3004 

3006 

3008 

200 

3010 

3012 

3015 

3017 

3019 

3021 

3023  3025 

3028 

3030 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

,519 


Table  II.    Values  and  Logarithms  of  Trigonometric  Functions 

[Characteristics  of  Logarithms  omitted  — determine  by  the  usual  rule  from  the  value] 


RADIANS 

DEGREES 

SINE 

TANGENT 

COTANGENT 

COSINE 

Value  Log10 

Value  Lo<r10 

Value   Logio 

Value   Log10 

.0000 

0°00' 

0000 

0000 

1.0000  .0000 

90°  00' 

1  .5708 

!(X)29 

10 

.0029  .4637 

.0029  .4637 

343.77  .5363 

1.0000  .0000 

50 

1^5679 

.0058 

20 

.0058  .7(548 

.0058  .7648 

171.89  .2352 

1.0000  .0000 

40 

1.5650 

.0087 

30 

.0087  .9408 

.0087  .9409 

114.59  .0591 

1.0000  .0000 

30 

1.5621 

.0116 

40 

.0116  .0658 

.0116  .0658 

85.940  .9342 

.9999  .0000 

20 

1.5592 

.0145 

50 

.0145  .1627 

.0145  .1627 

68.750  .8373 

.9999  .0000 

10 

1.5563 

.0175 

1°00' 

.0175  .2419 

.0175  .2419 

57.290  .7581 

.9998  .9999 

89°  00' 

1.5533 

.0204 

10 

.0204  .3088 

.0204  .3089 

49.104  .6911 

.9998  .9999 

50 

1.5504 

.0233 

20 

.0233  .3668 

.0233  .3(569 

42.964  .6331 

.9997  .9999 

40 

1.5475 

.02(52 

30 

.0262  .4179 

.0262  .4181 

38.188  .5819 

.9997  .9999 

30 

1.5446 

.021)1 

40 

.0291  .4637 

.0291  .4638 

34.368  .5362 

.9996  .9998 

20 

1.5417 

.0320 

50 

.0320  .5050 

.0320  .5053 

31.242  .4947 

.9995  .9998 

10 

1.5388 

.0349 

2°  00' 

.0349  .5428 

.0349  .5431 

28.636  .4569 

.9994  .9997 

8  8°  00' 

1.5359 

.0378 

10 

.0378  .5776 

.0378  .5779 

2(5.432  .4221 

.9993  .9997 

50 

1.5330 

.0407 

20 

.0407  .6097 

.0407  .6101 

24.542  .3899 

.9992  .9996 

40 

1.5301 

.0436 

30 

.0436  .6397 

.0437  .6401 

22.904  .3595) 

.9990  .9996 

30 

1.5272 

.0465 

40 

.0465  .6677 

.0466  .6682 

21.470  .3318 

.9989  .9995 

20 

1.5243 

.0495 

50 

.0494  .6940 

.0495  .6945 

20.206  .3055 

.9988  .9995 

10 

1.5213 

.0524 

3°  00' 

.0523  .7188 

.0524  .7194 

19.081  .2806 

.9986  .9994 

87°  00' 

1.5184 

.0553 

10 

.0552  .7423 

.0553  .7429 

18.075  .2571 

.9985  .C993 

50 

1.5155 

.0582 

20 

.0581  .7645 

.0582  .7652 

17.169  .2348 

.9983  .9993 

40 

1.5126 

.0611 

30 

.0610  .7857 

.0612  .7865 

16.350  .2135 

.9981  .9992 

30 

1.5097 

.0640 

40 

.0640  .8059 

.0(541  .8067 

15.605  .1933 

.9980  .9991 

20 

1.5068 

.0669 

50 

.0669  .8251 

.0670  .8261 

14.924  .1739 

.9978  .9990 

10 

1.5039 

.0698 

4°  00' 

.0698  .8436 

.0699  .8446 

14.301  .1554 

.9976  .9989 

86°  00' 

1.5010 

.0727 

10 

.0727  .8613 

.0729  .8624 

13.727  .1376 

.9974  .9989 

50 

1.4981 

.0756 

20 

.0756  .8783 

.0758  .8795 

13.197  .1205 

.9971  .9988 

40 

1.4952 

.0785 

30 

.0785  .8946 

.0787  .8960 

12.706  .1040 

.9969  .9987 

30 

1.4923 

.0814 

40 

.0814  .9104 

.0816  .9118 

12.251  .0882 

.9967  .9986 

20 

1.4893 

.0844 

50 

.0843  .9256 

.0846  .9272 

11.826  .0728 

.9964  .9985 

10 

1.4864 

.0873 

5°  00' 

.0872  .9403 

.0875  .9420 

11.430  .0580 

.9962  .9983 

85°  00' 

1.4835 

.0902 

10 

.0901  .9545 

.0904  .9563 

11.059  .0437 

.9959  .9982 

50 

1.4806 

.0931 

20 

.0929  .9682 

.0934  .9701 

10.712  .0299 

.9957  .9981 

40 

1.4777 

.09(50 

30 

.0958  .9816 

.0963  .9836 

10.385  .0164 

.9954  .9980 

30 

1.4748 

.0989 

40 

.0987  .9945 

.0992  .95X56 

10.078  .0034 

.9951  .9979 

20 

1.4719 

.1018 

50 

.1016  .0070 

.1022  .0093 

9.7882  .9907 

.9948  .9977 

10 

1.4690 

.1047 

6°  00' 

.1045  .0192 

.1051  .0216 

9.5144  .9784 

'.9945  .9976 

84°  00' 

1.4661 

.1076 

10 

.1074  .0311 

.1080  .0336 

9.'_'5.r)3  .9664 

.9942  .9975 

50 

1.4632 

.1105 

20 

.1103  .0426 

.1110  .0453 

9.0098  .9547 

.9939  .9973 

40 

1.4(503 

.1134 

30 

.1132  .0539 

.1139  .05(57 

8.7769  .94a3 

.9936  .9972 

30 

1.4573 

.1164 

40 

.1161  .0648 

.1169  .0678 

8.5555  .9322 

.9932  .9971 

20 

1.4544 

.1193 

50 

.1190  .0755 

.1198  .0786 

8.3450  .9214 

.9929  .9969 

10 

1.4515 

.1222 

7°  00' 

.1219  .0859 

.1228  .0891 

8.1443  .9109 

.9925  .9968 

83°  00' 

1.4486 

.1251 

10 

.1248  .0961 

.1257  .0995 

7.9530  .9005 

.9922  .9966 

50 

1.4457 

.1280 

20 

.1276  .10(50 

.1287  .109(5 

7.7704  .8904 

.9918  .9964 

40 

1.4428 

.1309 

30 

.1305  .1157 

.1317  .1194 

7.5958  .8806 

.9914  .9963 

30 

1.4399 

.1338 

40 

.1334  .1252 

.1346  .1291 

7.4287  .8709 

.9911  .9961 

20 

1.4370 

.1367 

50 

.13(53  .1345 

.1376  .1385 

7.2(587  .8615 

.9907  .9959 

10 

1.4341 

.1396 

8°  00' 

.1392  .1436 

.1405  .1478 

7.1154  .8522 

.9903  .9958 

82°  00' 

1.4312 

.1425 

10 

.1421  .1525 

.1435  .15(59 

6.9682  .8431 

.9899  .9956 

50 

1.4283 

.1454 

20 

.1449  .1612 

.1465  .1(558 

6.8269  .8342 

.9894  .9954 

40 

1.4254 

.1484 

30 

.1478  .1697 

.1495  .1745 

6.6912  .8255 

.9890  .9952 

30 

1.42-24 

.1513 

40 

.1507  .1781 

.1524  .1831 

6.5606  .8169 

.9886  .9950 

20 

1.4106 

.1542 

50 

.1536  .1863 

.1554  .1915 

6.4348  .8085 

.9881  .9948 

10 

1.41(56 

.1571 

9°  00' 

.1564  .1943 

.1584  .1997 

6.3138  .8003 

.9877  .9946 

81°  00' 

1.41  ."7 

Value  Log10 

Value  Lojrln 

Value   Log10 

Value   Log10 

DEGREES 

RADIANS 

COSINE 

COTANGENT 

TANGENT 

SINE 

320 


Table  II.     Values  and  Logarithms  of  Trigonometric  Functions 

[Characteristics  of  Logarithms  omitted  —determine  by  the  usual  rule  from  the  value] 


UADIANS 

DEGREES 

SINE 

TANGEN-T 

COTANGENT 

COSINE 

Value  Log10 

Value  Loi^m 

Value   L<>£10 

Value  Logla 

.1571 

9°  00' 

.1504  .1943 

.1584  .1997 

6.3138  .8003 

.9877  .9946 

81°  00' 

1.4137 

.1600 

10 

.1593  .2022 

.1014  .2078 

0.1970  .7922 

.9872  .9944 

50 

1.4108 

.1629 

20 

.1622  .2100 

.1044  .21.78 

6.0844  .7842 

.98(58  .9942 

40 

1.4079 

.1658 

30 

.1050  .2170 

.1673  .22:50 

5.9758  .77(54 

.9863  .9940 

30 

1.4050 

.1687 

40 

.1679  .22.-.  1 

.1703  .2313 

5.8708  .7(587 

.9858  .9938 

20 

1  .4021 

.1710 

50 

.1708  .2324 

.1733  .2389 

5.7094  .7611 

.9853  .99:1(5 

10 

1.3992 

.1745 

10°  00' 

.1736  .2397 

.1763  .2463 

5.6713  .7537 

.9848  .9934 

80°  00' 

1.39(53 

.1774 

10 

.1765  .2468 

.1793  .25:  JO 

5.5704  .7464 

.9843  .99:  :i 

50 

1.3934 

.1804 

20 

.1794  .25:38 

.1823  .2(509 

5.4845  .7391 

.9838  .9!  129 

40 

1.3904 

.1833 

30 

.1822  .2606 

.1853  .2(580 

f).:;9.-)5  .7320 

.9833  .9927 

30 

1.3875 

.1862 

40 

.1851  .2674 

.1883  .2750 

5.3093  .7250 

.9827  .9924 

20 

1.384fi 

.1891 

50 

.1880  .2740 

.1914  .2819 

5.2257  .7181 

.9822  .9922 

10 

1.3817 

.1920 

11°00' 

.1908  .2806 

.1944  .2887 

5.1446  .7113 

.9816  .9919 

79°  00' 

1.3788 

.1949 

10 

.1937  .2870 

.1974  .2!r,:; 

5.0658  .7047 

.9811  .9917 

50 

1.3759 

.1978 

20 

.1965  .2934 

.2004  .3020 

4.9894  .6980 

.9S05  .9914 

40 

1.3730 

5007 

30 

.1994  .2997 

.2035  .3085 

4.9152  .(5915 

.9799  .9912 

30 

1.3701 

.2036 

40 

.2022  .3058 

.20(55  .3149 

4.8430  .6851 

.9793  .9909 

20 

1.3672 

.2005 

50 

.2051  .3119 

.2095  .3212 

4.7729  .6788 

.9787  .9907 

10 

1.3643 

.2094 

12°  00' 

.2079  .3179 

.2126  .3275 

4.7046  .6725 

.9781  .9904 

78°  00' 

1.3614 

.'21--':$ 

10 

.2108  .:52:;s 

.2150  .3336 

4.6382  .6664 

.9775  .9901 

50 

1.3584 

5153 

20 

.2136  .3290 

.2186  .3397 

4.5736  .6(503 

.9769  .9899 

40 

1.3555 

.2182 

30 

.2164  .3353 

.2217  .3458 

4.5107  .6542 

.9763  .9896 

30 

1.3526 

.2211 

40 

.2193  .3410 

.2247  .3517 

4.4494  .0483 

.9757  .9893 

20 

1.3497 

.21-40 

50 

.2221  .3466 

.2278  .357(5 

4.3897  .6424 

.9750  .9890 

10 

1.3468 

5269 

13°  00' 

.2250  .3521 

.2309  .3634 

4.3315  .6366 

.9744  .9887 

77°  00' 

1.3439 

5298 

10 

.2278  .3575 

.2339  .3091 

4.2747  .0:509 

.97:57  .9884 

50 

1.3410 

.2327 

20 

.2306  .3629 

.2370  .3748 

4.2193  .02.72 

.9730  .'.issi 

40 

1.3381 

.2356 

30 

.2334  .30H2 

.2401  .3804 

4.1053  .6196 

.9724  .9878 

30 

1.3352 

5380 

40 

.2:503  .3734 

.2432  .3859 

4.1120  .6141 

.9717  .9S7.- 

20 

i  .:;:;23 

.2414 

50 

.2391  .3786 

.2462  .3914 

4.0(511  .6086 

.9710  .9872 

10 

1.3294 

.2443 

14°  00' 

.2419  .3837 

.2493  .3908 

4.0108  .6032 

.9703  .98(59 

76°  00' 

1.32(55 

5473 

10 

.2447  .3887 

.2524  .4021 

3.9(517  .5979 

.9090  .'.'sro 

00 

L.323B 

5502 

20 

.2470  .3937 

.2555  .4074 

3.9130  .5926 

.<iOs9  .9863 

40 

1.3200 

.2331 

30 

.2504  .3986 

.2586  .4127 

3.81  5G7  .5873 

.9681  .9S59 

30 

1.3177 

5560 

40 

.2532  .4035 

.2617  .4178 

3.8208  .5822 

.9674  .98.-.0 

20 

1.31  48 

.2589 

50 

.2560  .4083 

.2648  .4230 

3.7700  .5770 

.9007  .9853 

10 

1.3119 

.2618 

15°00' 

.2588  .4130 

.2679  .4281 

3.7321  .5719 

.9(559  .9849 

75  00' 

1.3090 

.2647 

10 

.2616  .4177 

.2711  .4331 

3.6891  .5(569 

.9052  .'.MI; 

no 

1.3061 

5676 

20 

.2641  .4223 

.2712  .4381 

3.0470  .5019 

.9(544  .984:5 

40 

1.3032 

5708 

30 

.2672  .420!) 

.2773  .4430 

30059  .5570 

.9(5:50 

30 

1.3003 

.2734 

40 

.2700  .4314 

.2805  .417!) 

3.5050  .5521 

.902S  .'.is:,-; 

20 

1.2974 

.2763 

50 

.2728  .4:»9 

.2836  .4527 

3.5261  .5473 

.9621  .9S32 

10 

15948 

5793 

16°  00' 

.2756  .4403 

.2S07  .4575 

3.4874  .5125 

.9613  .9-S2S 

74°  00' 

1.2918 

5822 

10 

.2784  .4447 

.2*99  .4(522 

3.4495  ..T.J7S 

.9605  .9*2". 

60 

.2851 

20 

.2812  .4491 

.29:51  .4(509 

3.1124  .5331 

.959(5  .9821 

40 

1.2857 

.2880 

30 

.2840  .4533 

.2902  .471(5 

3.3759  .52S4 

.!i:,ss  .9817 

30 

1.2828 

.2909 

40 

.2868  .4571! 

.2994  .4762 

3.3402  ..72:  IS 

.95SO  .9814 

20 

1.2799 

.2938 

50 

.2896  .4618 

.3026  .4808 

3.3052  .5192 

.9572  .9810 

10 

1.2770 

.2967 

17°  00' 

.2924  .4659 

.3057  .48.-,:? 

3.2709  .r,117 

.9503  .9SO<; 

73C00' 

1.2741 

5996 

10 

,2!i:,2  .4700 

.3089  .4S9S 

3.2371  .5102 

.'.).-,.-,.-,  .9M>2 

50 

1.2712 

.3028 

20 

.297!)  .4741 

..-5121  .4943 

3.2041  .r,0.-,7 

.9.140  .9798 

40 

1.2683 

.3054 

30 

.3007  .4781 

.3153  .4987 

3.1  7  10  ..Vi  115 

.9537  .97d! 

30 

1.2664 

.3083 

40 

.:;(»:•,;,  .4821 

.3185  .5():  ;i 

3.1:197  .4909 

.9528  .9790 

20 

1.2025 

.3113 

50 

.3062  .4861 

.3217  .5075 

3.1084  .4925 

.9520  .978(5 

10 

1.2595 

.3142 

18°  00' 

.3090  .4900 

.:-.2i9  ..-11* 

3.0777  .4882 

.9511  .97S2 

72°  00' 

1.2500 

Value   Logu 

V  III  Ui-    I.l'lTi,, 

Value   Loffio 

Value   I."L',n 

DEGREKB 

UADIANS 

Oomra 

COTANGENT 

|'AXI;ENT 

SINK 

321 


Table  II.     Values  and  Logarithms  of  Trigonometric  Functions 

[Characteristics  of  Logarithms  oniitii-d  — ilctcnnim'  !>y  the  usual  rule  IVom  tlic  valucj 


RADIANS 

DEGREES 

SINK 

TANGENT 

COTANGENT 

COSINE 

Value  Log10 

Value  Log10 

Value   Log-10 

Value   Lopr10 

.3142 

18°  00' 

.3090  .4900 

.3249  .5118 

3.0777  .4882 

.9511  .9782 

72°  00' 

1.2500 

.3171 

10 

.3118  .4939 

.3281  .5101 

3.0475  .4839 

.9502  .9778 

50 

1.2537 

.3200 

20 

.3145  .4977 

.3314  .5203 

3.0178  .4797 

.9492  .9774 

40 

1.2508 

.3221  i 

30 

.3173  .5015 

.3340  .5245 

2.9887  .4755 

.9483  .9770 

30 

1.2479 

.3258 

40 

.3201  .5052 

.3378  .5287 

2.9(500  .4713 

.9474  .9705 

20 

1.2450 

.3287 

50 

.3228  .5090 

.3411  .5329 

2.9319  .4671 

.9465  .97(51 

10 

1.2421 

.3316 

19°  00' 

.3250  .5120 

.3443  .5370 

2.9042  .4030 

.9455  .9757 

71°  00' 

1.2392 

.3345 

10 

.3283  .5103 

.3476  .5411 

2.8770  .4589 

.944(5  .9752 

50 

1.2363 

.3374 

20 

.3311  .5199 

.3508  .5451 

2.8502  .4549 

.943(5  .9748 

40 

1.2334 

.3403 

30 

.3338  .5235 

.3541  .5491 

2.8239  .4509 

.9420  .9743 

30 

1.23!  '5 

.3432 

40 

.33<55  .5270 

.3574  .5531 

2.7980  .4409 

.9417  .9739 

20 

1.2275 

.341  W 

50 

.3393  .5300 

.3007  .5571 

2.7725  .4429 

.9407  .9734 

10 

1.22-tfi 

.3401 

20°  00' 

.3420  .5341 

.3(540  .5011 

2.7475  .4389 

.9397  .9730 

70°  00' 

1  .2217 

5520 

10 

.3448  .5375 

.3073  .5050 

2.7228  .4350 

.9387  .9725 

50 

1.2188 

.3649 

20 

.3475  .5409 

.370(5  .5089 

2.6985  .4311 

.9377  .9721 

40 

1.2150 

.3578 

30 

.3502  .5443 

.3739  .5727 

2.0740  .4273 

.9367  .9716 

30 

1.2130 

.3007 

40 

.3529  .5477 

.3772  .5766 

2.0511  .4234 

.9350  .9711 

20 

1.2101 

.3030 

50 

.3557  .5510 

.3805  .5804 

2.6279  .419(5 

.9340  .9700 

10 

1.2072 

.3005 

21°  00' 

..3584  .5543 

.3839  .5842 

2.6051  .4158 

.9336  ,9702 

69°  00' 

1.2043 

.3694 

10 

.3011  .5570 

.3872  .5879 

2.5826  .4121 

.9325  .9097 

50 

1.2014 

.3723 

20 

.3(538  .5009 

.3900  .5917 

2.5(505  .4083 

.9315  .9092 

40 

1.1985 

.3752 

30 

.3I505  .5041 

.3939  .5954 

2.5386  .4046 

.9304  .9087 

30 

1.1950 

.3782 

40 

.3092  .5(573 

.3973  .5991 

2.5172  .4009 

.9293  .9(582 

20 

1.1920 

.3811 

50 

.3719  .5704 

.4000  .0028 

2.4960  .3972 

.9283  .1)077 

10 

1.1897 

.3840 

22°  00' 

.3746  .5730 

.4040  .6064 

2.4751  .3936 

.9272  .9(572 

68°  00' 

1.1808 

.3860 

10 

.3773  .5707 

.4074  .6100 

2.4545  .3900 

.9201  .1X507 

50 

1.1839 

.3808 

20 

.3800  .5798 

.4108  .0130 

2.4342  .38(54 

.9250  .9001 

40 

1.1810 

.3027 

30 

.3827  .5828 

.4142  .6172 

2.4142  .3828 

.9239  .1)050 

30 

1.1781 

.35)66 

40 

.3854  .5859 

.4176  .6208 

2.3945  .3792 

.9228  .9651 

20 

1.1752 

.3983 

50 

.3881  .5889 

.4210  .6243 

2.3750  .3757 

.9216  .9046 

10 

1.1723 

.4014 

23°  00' 

.3907  .5919 

.4245  .6279 

2.3559  .3721 

.9205  .9640 

67°  00' 

1.1094 

.4043 

10 

.3934  .5948 

.4279  .6314 

2.3309  .3086 

.9194  .9(535 

50 

I.IK;.-, 

.4072 

20 

.3901  .5978 

.4314  .6348 

2.3183  .3652 

.9182  .9629 

40 

1.10.30 

.4102 

30 

.3987  .0007 

.4348  .6383 

2.2998  .3(517 

.9171  .9024 

30 

1.1000 

.4131 

40 

.4014  .0036 

.4383  .6417 

2.2817  .3583 

.9159  .9018 

20 

1.1577 

.41(50 

50 

.4041  .0005 

.4417  .6452 

2.2637  .3548 

.W147  .9013 

10 

1.1548 

.4189 

24°  00' 

.4067  .0093 

.4452  .6486 

2.24(50  .3514 

.9135  .9007 

66°  00' 

1.1519 

.4218 

10 

.4094  .6121 

.4487  .0520 

2.2286  .3480 

.9124  .9602 

50 

1.1400 

.4247 

20 

.4120  .6149 

.4522  .(5553 

2.2113  .3447 

.9112  .9596 

40 

1.1401 

.4270 

30 

.4147  .0177 

.4557  .6587 

2.1943  .3413 

.9100  .9590 

30 

1.1432 

.4300 

40 

.4173  .6205 

.4592  .6620 

2.1775  .3380 

.9088  .9584 

20 

1.1403 

.4334 

50 

.4200  .6232 

.4628  .0654 

2.1609  .3346 

.9075  .9579 

10 

1.1374 

.4303 

25°  00' 

.4220  .0259 

.4063  .6687 

2.1445  .3313 

.9063  .9573 

65°  00' 

1.1346 

.4392 

10 

.4253  .6280 

.4699  .0720 

2.1283  .3280 

.9051  .95(57 

50 

1.1310 

.4422 

20 

.4279  .6313 

.4734  .0752 

2.1123  .3248 

.9038  .95(51 

40 

1.128H 

.4451 

30 

.4305  .6340 

.4770  .0785 

2.09(55  .3215 

.9020  .9555 

30 

1.1257 

.4480 

40 

.4331  .6:366 

.4800  .0817 

2.0809  .3183 

.9013  .9549 

20 

1.1228 

.4501) 

50 

.4358  .0392 

.4841  .0850 

2.0(555  .3150 

.9001  .9543 

10 

1.1190 

.4538 

26°  00' 

.4384  .6418 

.4877  .0882 

2.0503  .3118 

.8988  .9537 

64°  00' 

1.1170 

.4507 

10 

.4410  .6444 

.4913  .0914 

2.0353  .3080 

.8975  .95:50 

50 

1.1141 

.4590 

20 

.4436  .6470 

.4950  .0940 

2.0204  .3054 

.89(52  .9524 

40 

1.1112 

.4(525 

30 

.4402  .6495 

.4986  .15977 

2.0057  .3023 

.8949  .9518 

30 

1.1083 

.4054 

40 

.4488  .6521 

.5022  .7009 

1.9912  .21)91 

.8930  .9512 

20 

1.1054 

.4(583 

50 

.4514  .6546 

.5059  .7040 

1.9768  .29(50 

.8923  .9505 

10 

1.1025 

.4712 

27°  00' 

.4540  .6570 

.5095  .7072 

1.9026  .2928 

.8910  .9499 

63°  00' 

1.0996 

Value  Logjo 

Value  Login 

Value   Log10 

Value   Logjo 

DEGREES 

RADIANS 

COSINE 

COTANGENT 

TANGENT 

SINE 

322 


Table  II.     Values  and  Logarithms  of  Trigonometric  Functions 

[Characteristics  of  Logarithms  omitted — determine  by  the  usual  rule  from  the  value] 


RADIANS 

DEGREEK 

SlXE 

TANGENT 

COTANGENT 

COSINE 

Value  Log10 

Value  Log10 

Value   Log10 

Value  Log10 

.4712 

27°  00' 

.4540  .6570 

.5095  .7072 

1.9(526  .2928 

.8910  .9499 

63°  00' 

1.0996 

.4741 

10 

.45(56  .6595 

.5132  .7103 

1.9486  .2897 

.8897  .9492 

50 

1.09(5(5 

.4771 

20 

.4592  .6620 

.5169  .7134 

1.9347  .28(56 

.8884  .948(5 

40 

1.0937 

.4800 

30 

.4617  .6644 

.5206  .7165 

1.9210  .2835 

.8870  .9479 

30 

1.0908 

.4829 

40 

.4643  .66(58 

.5243  .7196 

1.9074  .2804 

.8857  .9473 

20 

1.0879 

.4858 

50 

.4669  .6692 

.5280  .7226 

1.8940  .2774 

.8843  .9466 

10 

1.0850 

.4887 

28°  00' 

.4695  .6716 

.5317  .7257 

1.8807  .2743 

.8829  .9459 

62°  00' 

1.0821 

.4916 

10 

.4720  .(5740 

.5354  .7287 

1.8676  .2713 

.8816  .9453 

50 

1.0792 

.4945 

20 

.4746  .6763 

.5392  .7317 

1.8546  .2683 

.8802  .944(5 

40 

1.07(53 

.4974 

30 

.4772  .6787 

.5430  .7348 

1.8418  .2652 

.8788  .9439 

30 

1.0734 

.5003 

40 

.4797  .6810 

.5467  .7378 

1.8291  .2(522 

.8774  .9432 

20 

1.0705 

.5032 

50 

.4823  .6833 

.5505  .7408 

1.8165  .2592 

.8760  .9425 

10 

1.0676 

.5061 

29°  00' 

.4848  .6856 

.5543  .7438 

1.8040  .2562 

.8746  .9418 

61°  00' 

1.0647 

.5091 

10 

.4874  .6878 

.5581  .7467 

1.7917  .2533 

.8752  .9411 

50 

1.  0(517 

.5120 

20 

.4899  .6901 

.5619  .7497 

1.7796  .2503 

.8718  .9404 

40 

1.05S8 

.5149 

30 

.4924  .6923 

.5658  .7526 

1.7675  .2474 

.8704  .9397 

30 

1.0559 

.5178 

40 

.4950  .6946 

.6696  .7556 

1.7556  .2444 

.8689  .9390 

20 

1.0530 

.5207 

50 

.4975  .6968 

.5735  .7585 

1.7437  .2415 

.8675  .9383 

10 

1.0501 

.5236 

30°  00' 

.5000  .6990 

.5774  .7614 

1.7321  .2386 

.8660  .9375 

60°  00' 

1.0472 

.5265 

10 

.5025  .7012 

.5812  .7644 

1.7205  .2356 

.8646  .9368 

50 

1.0443 

.521)4 

20 

.5050  .7033 

.5851  .7673 

1.7090  .2327 

.8631  .9361 

40 

1.0414 

.5323 

30 

.5075  .7055 

.5890  .7701 

1.6977  i2299 

.8(516  .9353 

30 

1.0385 

.6352 

40 

.5100  .7076 

.5930  .7730 

1.6864  .2270 

.8601  .9346 

20 

1.0356 

.5381 

50 

.5125  .7097 

.5969  .7759 

1.6753  .2241 

.8587  .9338 

10 

1.0327 

.5411 

31°  00' 

.5150  .7118 

.6009  .7788 

1.6643  .2212 

.8572  .9331 

59°  00' 

1.0297 

.5440 

10 

.5175  .7139 

.6048  .781(5 

1.6534  .2184 

-S557  .9323 

50 

1.0268 

.5469 

20 

.5200  .7160 

.6088  .7845 

1.6426  .2155 

.8542  .9:;i5 

40 

1.0239 

.5498 

30 

.5225  .7181 

.6128  .7873 

1.6319  .2127 

.8526  .9308 

30 

1.0210 

.5527 

40 

.5250  .7201 

.6168  .7902 

1.6212  .2098 

.8511  .9:500 

20 

1.0181 

.5556 

50 

.5275  .7222 

.6208  .7930 

1.6107  .2070 

.8496  .9292 

10 

1.0152 

.5585 

32°  00' 

.5299  .7242 

.6249  .7958 

1.6003  .2042 

.8480  .9284 

58°  00' 

1.0123 

.5(514 

10 

.5324  .7262 

.6289  .798(5 

1.5900  .2014 

.84(55  .927(5 

50 

1.0094 

.5643 

20 

.5348  .7282 

.6330  .8014 

1.5798  .19S6 

.8450  .9268 

40 

1.0065 

.5(572 

30 

.5373  .7302 

.6371  .8042 

1.5697  .1958 

.8434  .9260 

30 

1.003(5 

.5701 

40 

.5398  .7322 

.6412  .8070 

1.55!  )7  .1930 

.8418  .9252 

20 

1.0007 

.5730 

50 

.5422  .7342 

.6453  .8097 

1.5497  .1903 

.8403  .9244 

10 

.9977 

.5760 

33°  00' 

.5446  .7361 

.6494  .8125 

1.5399  .1875 

.8387  .92:!6 

57°  00' 

.9948 

0789 

10 

.5471  .7380 

.(55.'56  .8153 

1.5301  .1847 

.8371  .9228 

50 

.9919 

.5818 

20 

.5495  .7400 

.6577  .8180 

1.5204  .1820 

.8.'555  .9219 

40 

.9890 

.5847 

30 

.5519  .7419 

.(',619  .8208 

1.5108  .1792 

.8339  .9211 

30 

.98(51 

.5876 

40 

.5544  .743S 

.6(561  .8235 

1.5013  .17C.5 

..s:  !23  .9203 

20 

.9632 

.5905 

50 

.5568  .7457 

.(5703  .8263 

1.4919  .1737 

.8307  .9194 

10 

.9803 

.5934 

34°  00' 

.5592  .7476 

.6745  .8290 

1.4826  .1710 

.8290  .918(5 

56°  00' 

.9774 

.5963 

10 

.5(516  .7494 

.6787  .8317 

1.4733  .1683 

.8274  .9177 

50 

.9745 

.5992 

20 

.5640  .7513 

.68:50  .8344 

1.4641  .1656 

.8258:.  91(19 

40 

.9716 

.6021 

30 

.5(5(54  .7531 

.6873  .8371 

1.4650  .1629 

.8241  .91(50 

30 

.'.H1S7 

.6050 

40 

.5<;ss  .7550 

.6916  .8398 

1.44(50  .1602 

.-S-J25  .9151 

20 

.9657 

.6080 

60 

.5712  .7568 

.6959  .8425 

1.4370  .1575 

.8208  .9142 

10 

.9628 

.6109 

35°  00' 

.5736  .7586 

.7002  .8452 

1.4281  .1548 

.8192  .9134 

55°  00' 

.9599 

.6138 

10 

.57(50  .7(504 

.Tdlil  .8479 

1.4193  .1521 

.8175  .9125 

50 

.9570 

.6167 

20 

.5783  .7622 

.7089  .8506 

1.4106  .1494 

.8158  .9116 

40 

.9541 

.6196 

30 

.5S07  .7640 

.7133  .S5.0,:; 

1.4019  .14(57 

.8141  .9107 

30 

.9512 

.6225 

40 

.5831  .7(157 

.7177  .8559 

1.3934  .1441 

.8124  .9098 

20 

.9483 

.6254 

5i  > 

.5854  .7675 

.7221  .8586 

1.3848  .1414 

.8107  .9089 

10 

.9454 

.6283 

36°  00' 

.5878  .7692 

.7265  .8613 

1.37(54  .1387 

.8090  .9080 

54°  00' 

.9425 

Value  Log10 

Value  Log10 

Value   Lop10 

Value   Log10 

D  EG  BEES 

RADIANS 

COSINB 

COTANGENT 

TANGENT 

SINE 

323 


Table  n.     Values  and  Logarithms  of  Trigonometric  Functions 

[Characteristics  of  Logarithms  omitted  —  determine  by  the  usual  rule  from  the  value] 


KAIHA  N 

DEGREE 

SINK 

TANGENT 

COTANGENT 

COSINE 

Value  Logj 

Value   LOJBTJ 

Value   Logle 

Value  Logt 

.6283 

36°  00 

.5878  .7692 

.7265  .8613 

1.3764  .1387 

.8090  .9080 

54°  00 

.9425 

.6312 

10 

.5901  .7710 

.7310  .8639 

1.3680  .1361 

.8073  .9070 

50 

.9396 

.6341 

20 

.5925  .7727 

.7355  .8666 

1.3597  .1334 

.8056  .9061 

40 

.9367 

.6370 

30 

.5948  .7744 

.7400  .8692 

1.3514  .1:308 

.8039  .91152 

30 

.9338 

.6400 

40 

.5972  .7761 

.7445  .8718 

1.3432  .1282 

.8021  .9042 

20 

.9308 

.6429 

50 

.5995  .7778 

.7490  .8745 

1.3351  .1255 

.8004  .9033 

10 

.9279 

.6458 

37°  00 

.6018  .7795 

.7536  .8771 

1.3270  .1229 

.7986  .9023 

53°  00 

.9250 

.6487 

10 

.6041  .7811 

.7581  .8797 

1.3190  .1203 

.7969  .9014 

50 

.9221 

.6516 

20 

.6065  .7828 

.7627  .8824 

1.3111  .1176 

.7951  .9004 

40 

.9192 

.6545 

30 

.6088  .7844 

.7673  .8850 

1.3032  .1150 

.7934  .8995 

30 

.9163 

.6574 

40 

.6111  .7861 

.7720  .8876 

1.2954  .1124 

.7916  .8985 

20 

.9i;34 

.6603 

50 

.6134  .7877 

.7766  .8902 

1.2876  .1098 

.7898  .8975 

10 

.9105 

.6632 

38°  00' 

.6157  .7893 

.7813  .8928 

1.2799  .1072 

.7880  .8965 

52°  00 

.8076 

.6661 

10 

.6180  .7910 

.7860  .8954 

1.2723  .1046 

.7862  .8955 

50 

.9047 

.6690 

20 

.6202  .7926 

.7907  .8980 

1.2647  .1020 

.7844  .8945 

40 

.9018 

.6720 

30 

.6225  .7941 

.7954  .9006 

1.2572  .0994 

.7826  .8935 

30 

.8988 

.6749 

40 

.6248  .7957 

.8002  .9032 

1.2497  .09(58 

.7808  .8925 

20 

.8959 

.6778 

50 

.6271  .7973 

.8050  .9058 

1.2423  .0942 

.7790  .8915 

10 

.8930 

.6807 

39°  00' 

.6293  .7989 

.8098  .9084 

1.2349  .0916 

.7771  .8905 

51°  00 

.8901 

.6836 

10 

.6316  .8004 

.8146  .9110 

1.2276  .0890 

.7753  .8895 

50 

.8872 

.6865 

20 

.6338  .8020 

.8195  .9135 

1.2203  .0865 

.7735  .8884 

40 

.8843 

.6894 

30 

.6361  .80:35 

.8243  .9161 

1.2131  .0839 

.7716  .8874 

30 

.8814 

.6923 

40 

.6383  .8050 

.8292  .9187 

1.205!)  .0813 

.7698  .8864 

20 

.8785 

.6952 

50 

.6406  .8066 

.8342  .9212 

1.1988  .0788 

.7679  .8853 

10 

.8756 

.6981 

40°  00' 

.6428  .8081 

.8391  .9238 

1.1918  .0762 

.7660  .8843 

50°  00 

.8727 

.7010 

10 

.6450  .8096 

.8441  .9264 

1.1847  .0736 

.7642  .8832 

60 

.8698 

.7039 

20 

.6472  .8111 

.8491  .9289 

1.1778  .0711 

.7623  .8821 

40 

.8668 

.7069 

30 

.6494  .8125 

.8541  .9315 

1.1708  .0685 

.7604  .8810 

30 

.8639 

.7098 

40 

.6517  .8140 

.8591  .9341 

1.1640  .0659 

.7585  .8800 

20 

.8610 

.7127 

50 

.6539  .8155 

.8642  .9366 

1.1571  .0634 

.7566  .8789 

10 

.8581 

.7156 

41°  00' 

.6561  .8169 

.8693  .9392 

1.1504  .0608 

.7547  .8778 

49°  00 

.8552 

.7185 

10 

.6583  .8184 

.8744  .9417 

1.1436  .0583 

.7528  .8767 

50 

.8523 

.7214 

20 

.6604  .8198 

.8796  .9443 

1.1369  .0557 

.7509  .8756 

40 

.841)4 

.7243 

30 

.6626  .8213 

.8847  .9468 

1.1303  .0532 

.7490  .8745 

30 

.8465 

.7272 

40 

.6648  .8227 

.8899  .9494 

1.1237  .0506 

.7470  .8733 

20 

.8436 

.7301 

50 

.6670  .8241 

.8952  .9519 

1.1171  .0481 

.7451  .8722 

10 

.8407 

.7330 

42°  00' 

.6691  .8255 

.9004  .9544 

1.1106  .0456 

.7431  .8711 

48°  00' 

.8378 

.7359 

10 

.6713  .8269 

.9057  .9570 

1.1041  .0430 

.7412  .86<)9 

50 

.8348 

.7389 

20 

.6734  .8283 

.9110  .9595 

1.0977  .0405 

.7392  .8688 

40 

.8319 

.7418 

30 

.6756  .8297 

.9163  .9621 

1.0913  .0379 

.7373  .8676 

30 

.8290 

.7447 

40 

.6777  .8311 

.9217  .9646 

1.0850  .0354 

.7353  .8665 

20 

.8261 

.7476 

50 

.6799  .8324 

.9271  .9671 

1.0786  .0329 

.7333  .8653 

10 

.8232 

.7505 

43°  00' 

.6820  .8338 

.9325  .9697 

1.0724  .0303 

.7314  .8641 

47°  00' 

.8203 

.7534 

10 

.6841  .8351 

.9380  .9722 

1.0661  .0278 

.7294  .8629 

50 

.8174 

.7563 

20 

.6862  .8365 

.9435  .9747 

1.059!)  .0253 

.7274  .8618 

40 

.8145 

.7592 

30 

.6884  .8378 

.9490  .9772 

1.0538  .0228 

.7254  .8606 

30 

.8116 

.7621 

40 

.6905  .8391 

.9545  .9798 

1.0477  .0202 

.7234  .8594 

20 

.8087 

.7650 

50 

.6926  .8405 

.9601  .9823 

1.0416  .0177 

.7214  .8582 

10 

.8058 

.7679 

44°  00' 

.6947  .8418 

.9657  .9848 

1.0355  .0152 

.7193  .8569 

46°  00' 

.8029 

.7709 

10 

.6967  .8431 

.9713  .9874 

1.0295  .0126 

.7173  .8557 

50 

.7999 

.7738 

20 

.6988  .8444 

.9770  .9899 

1.0235  .0101 

.7153  .8545 

40 

.7970 

.7767 

30 

.7009  .8457 

.9827  .9924 

1.0176  .0076 

.7133  .8532 

30 

.7941 

.7796 

40 

.7030  .8469 

.9884  .9949 

1.0117  .0051 

.7112  .8520 

20 

.7912 

.7825 

50 

.7050  .8482 

.9942  .9975 

1.0058  .0025 

.7092  .8507 

10 

.7883 

.7854 

45°  00' 

.7071  .8495 

1.0000  .0000 

1.0000  .0000 

.7071  .8495 

45°  00 

.7854 

Value  Log10 

Value   Login 

Value   Log]0 

Value  Log10 

)EGREES 

vADIANS 

COSINE 

COTANGENT 

TANGENT 

SINE 

324 


Table  III.     Radian  Measure — Trigonometric  Functions 


Had. 

Deg.  Mln. 

sin. 

cos. 

tan. 

Rad. 

Deg.  Mln. 

sin. 

cos. 

tan 

0.0 

0   0 

0 

1 

0 

3.2 

183  20.8 

-.058 

-.998 

.058 

0.1 

5  43.8 

.100 

.995 

.100 

3.3 

189   4.6 

-.158 

-.987 

.161 

0.2 

11  27.5 

.199 

.980 

.203 

3.4 

194  48.3 

-.255 

-.967 

.264 

0.3 

17  11.3 

.296 

.955 

.309 

3.5 

200  32.1 

-.351 

-.936 

.375 

0.4 

22  55.1 

.389 

.921 

.423 

3.6 

206  15.9 

-.443 

-.897 

.493 

0.5 

28  38.9 

.479 

.878 

.546 

3.7 

211  59.7 

-.530 

-.848 

.625 

0.6 

34  22.6 

.565 

.825 

.684 

3.8 

217  43.4 

-.612 

-.791 

.774 

0.7 

40   6.4 

.644 

.765 

.842 

3.9 

223  27.2 

-.688 

-.726 

.947 

0.8 

45  50.2 

.717 

.697 

1.030 

4.0 

229  11.0 

-.757 

-.654 

1.158 

0.9 

51  34.0 

.783 

.622 

1.260 

4.1 

234  54.8 

-.818 

-.575 

1.424 

1.0 

57  17.7 

.841 

.540 

1.557 

4.2 

240  38.5 

-.872 

-.490 

1.778 

1.1 

63   1.5 

.891 

.454 

1.965 

4.3 

246  22.3 

-.916 

-.401 

2.286 

1.2 

68  45.3 

.932 

.362 

2.572 

4.4 

252   6.1 

-.952 

-.307 

3.096 

1.3 

74  29.1 

.964 

.267 

3.602 

4.5 

257  49.9 

-.978 

-.211 

4.638 

1.4 

80  12.8 

.985 

.170 

5.798 

4.6 

263  33.6 

-.994 

-.112 

8.859 

1.5 

85  56.6 

.997 

.071 

14.101 

4.7 

269  17.4 

-1.00 

-.012 

80.713 

1.6 

91  40.4 

1.000 

-.029 

-34.233 

4.8 

275   1.2 

-.996 

.088 

-11.385 

1.7 

97  24.2 

.992 

-.129 

-  7.700 

4.9 

280  45.0 

-.982 

.187 

-  5.267 

1.8 

103   7.9 

.974 

-.227 

-  4.286 

5.0 

286  28.6 

-.959 

.284 

-  3.381 

1.9 

108  51.7 

.946 

-.323 

-  2.927 

5.1 

292  12.5 

-.926 

.378 

-  2.449 

2.0 

114  35.5 

.909 

-.416 

-  2.185 

5.2 

297  56.3 

-.883 

.469 

-  1.885 

2.1 

120  19.3 

.863 

-.505 

-  1.710 

5.3 

303  40.1 

-.832 

.554 

-  1.501 

2.2 

126   3.0 

.808 

-.588 

-  1.374 

5.4 

309  23.8 

-.773 

.635 

-  1.217 

2.3 

131  46.8 

.746 

-.666 

-  1.119 

5.5 

315   7.6 

-.706 

.709 

-  .996 

2.4 

137  30.6 

.675 

-.737 

.917 

5.6 

320  51.4 

-.631 

.776 

-  .814 

2.5 

143  14.4 

.598 

-.801 

.747 

5.7 

326  35.2 

-.551 

.835 

-  .600 

2.6 

148  58.1 

.516 

-.857 

-  .602 

5.8 

332  18.9 

-.465 

.886 

-  .525 

2.7 

154  41.9 

.427 

-.904 

-  .473 

5.9 

338   2.7 

-.374 

.927 

-  .403 

2.8 

160  25.7 

.335 

-.942 

-  .356 

6.0 

343  46.5 

-.279 

.960 

-  .291 

2.9 

166   9.5 

.239 

-.971 

-  .246 

6.1 

349  30.3 

-.182 

.983 

-  .185 

3.0 

171  53.2 

.141 

-.990 

.143 

6.2 

355  14.0 

-.083 

.997 

-  .083 

3.1 

177  37.0 

.042 

-.999 

.042 

6.3 

360  57.8 

+.017 

1.000 

+  .017 

325 


Table  IV.    Squares  and  Cubes    Square  Roots  and  Cube  Roots 


No. 

SQUARE 

C'UUE 

SQUARE 
UOOT 

CUBE 

ItOOT 

No. 

SQUARE 

CUBE 

SQUARE 
KOOT 

ClT.E 

HOOT 

1 

1 

1 

1.000 

1.000 

51 

2,601 

132,651 

7.141 

3.708 

2 

4 

8 

1.414 

1.260 

52 

2,704 

140,608 

7.211 

3.733 

3 

9 

27 

1.732 

1.442 

53 

2,809 

148,877 

7.280 

3.75(5 

4 

16 

64 

2.000 

1.587 

54 

2,916 

157,464 

7.348 

3.780 

5 

25 

125 

2.236 

1.710 

55 

3,025 

166,375 

7.416 

3.803 

6 

36 

216 

2.449 

1.817 

56 

3,136 

175,616 

7.483 

3.826 

7 

49 

343 

2.646 

1.913 

57 

3,249 

185,193 

7.550 

3.84!) 

8 

64 

512 

2.828 

2.000 

58 

3,364 

195,112 

7.616 

3.871 

9 

81 

729 

3.000 

2.080 

59 

3,481 

205,379 

7.681 

3.893 

10 

100 

1,000 

3.162 

2.154 

60 

3,600 

216,000 

7.746 

3.915 

11 

121 

1,331 

3.317 

2.224 

61 

3,721 

226,981 

7.810 

3.936 

12 

144 

1,728 

3.464 

2.289 

62 

3,844 

238,328 

7.874 

3.968 

13 

109 

2,197 

3.606 

2.351 

63 

3,969 

250,047 

7.937 

3.979 

14 

196 

2,744 

3.742 

2.410 

64 

4,096 

262,144 

8.000 

4.000 

15 

225 

3,375 

3.873 

2.466 

65 

4,225 

274,625 

8.062 

4.021 

16 

256 

4,096 

4.000 

2.520 

66 

4,356 

287,496 

8.124 

4.041 

17 

.289 

4,913 

4.123 

2.571 

67 

4,489 

300,763 

8.185 

4.062 

18 

324 

5,832 

4.243 

2.621 

68 

4,624 

314,432 

8.246 

4.082 

19 

361 

6,859 

4.359 

2.668 

69 

4,761 

328,509 

8.307 

4.102 

20 

400 

8,000 

4.472 

2.714 

70 

4,900 

343,000 

8.367 

4.121 

21 

441 

9,261 

4.583 

2.759 

71 

5,041 

357,911 

8.426 

4.141 

22 

484 

10,648 

4.690 

2.802 

72 

5,184 

373,248 

8.485 

4.160 

23 

529 

12,167 

4.796 

2.844 

73 

5,329 

389,017 

8.544 

4.17H 

24 

576 

13,824 

4.899 

2.884 

74 

5,476 

405,224 

8.602 

4.198 

25 

625 

15,625 

5.000 

2.924 

75 

5,625 

421,875 

8.660 

4.217 

26 

676 

17,576 

5.099 

2.962 

76 

5,776 

438,!>76 

8.718 

4.236 

27 

729 

19,683 

5.196 

3.000 

77 

5,929 

456,533 

8.775 

4.254 

28 

784 

21,952 

5.292 

3.037 

78 

6,084 

474,552 

8.832 

4.273 

29 

841 

24,389 

5.385 

3.072 

79 

6,241 

493,039 

8.888 

4.291 

30 

900 

27,000 

5.477 

3.107 

80 

6,400 

512,000 

8.944 

4.:50!> 

31 

961 

29,791 

5.568 

3.141 

81 

6,561 

531,441 

9.000 

4.  :','21 

32 

1,024 

32,768 

5.657 

3.175 

82 

6,724 

551,368 

9.055 

4.:U4 

33 

1,089 

35,937 

5.745 

3.208 

83 

6,889 

571,787 

9.110 

4.3(52 

34 

1,156 

39,304 

5.831 

3.240 

84 

7,056 

592,704 

9.165 

4.380 

35 

1,225 

42,875 

5.916 

3.271 

85 

7,225 

614,125 

9.220 

4.397 

36 

1,2'W 

46,656 

6.000 

3.302 

86 

7,396 

636,056 

9.274 

4.414 

37 

1,3(59 

50,653 

6.083 

3.332 

87 

7,569 

658,503 

9.327 

4.431 

38 

1,444 

54,872 

6.164 

3.362 

88 

7,744 

681,472 

9.381 

4.448 

39 

1,521 

59,319 

6.245 

3.391 

89 

7,921 

704,969 

9.434 

4.4(;r, 

40 

1,600 

64,000 

6.325 

3.420 

90 

8,100 

729,000 

9.487 

4.481 

41 

1,681 

68,921 

6.403 

3.448 

91 

8,281 

753,671 

9.588 

4.49S 

42 

1,764 

74,088 

6.481 

3.476 

92 

8,464 

778,688 

9.592 

4.614 

43 

1,849 

79,507 

6.557 

3.503 

93 

8,649 

804,a57 

9.644 

4.631 

44 

1,<>:!6 

85,184 

6.633 

3.530 

94 

8,836 

HlJu,.-^ 

9.698 

4.547 

45 

2,025 

91,125 

6.708 

3.557 

95 

9,025 

857,370 

!>.747 

4.56.", 

46 

2,116 

97,336 

6.782 

3.583 

96 

9,216 

884,736 

9.798 

4.579 

47 

2,20!» 

103,823 

6.856 

3.609 

97 

9,409 

912,673 

9.849 

4.595 

48 

2,304 

110,592 

6.  928 

3.634 

98 

9,604 

941,192 

9.899 

4.610 

49 

2.401 

117,649 

7.000 

3.<>59 

99 

9,801 

970,299 

9.950 

4.626 

50 

2,500 

125,000 

7.071 

3.684 

100 

10,000 

1,000,000 

10.000 

4.1142 

For  a  more  complete  table,  see  THE  MACMILLAN  TABLES,  pp.  94-111. 


326 


Table  V.     Logarithms  of  Important  Constants 


Ar  =  NUMBER 

VALUE  OF  JV 

LOGjo  ff 

f 

3.14159265 

0.49714987 

1     +     * 

0.31830989 

9.50285013 

X* 

9.86960440 

0.99429975 

V^r 

1.77245385 

0.24857494 

e  =  Napierian  Base 

2.71828183 

0.43429448 

M  =  logio  e 

0.43429448 

9.63778431 

1  -H  M  =  loge  10 

2.30258509 

0.36221569 

!     180  -i-  r  =  degrees  in  1  radian 

57.2957795 

1.75812262 

IT  -=-  180  =  radians  in  1° 

0.01745329 

8.24187738 

IT  -T-  10800  =  radians  in  1' 

0.0002908882 

6.4637261 

x  H-  648000  =  radians  in  1" 

0.000004848136811095 

4.68557487 

sin  1" 

0.000004848136811076 

4.68557487 

tan  1" 

0.000004848136811152 

4.68557487 

centimeters  in  1  ft. 

30.480 

1.4840158 

feet  in  1  cm. 

0.032808 

8.5159842 

inches  in  1  m. 

39.37 

1.5951654 

pounds  in  1  kg. 

2.20462 

0.3433340 

kilograms  in  1  Ib. 

0.453593 

9.6566660 

g 

32.16  ft.  /sec.  /sec. 

1.5073160 

=  981  cm.  /sec.  /sec. 

2.9916690 

weight  of  1  cu.  ft.  of  water 

62.425  Ib.  (max.  density) 

1.7953586 

weight  of  1  cu.  ft.  of  air 

0.0807  Ib.  (at  32°  F.) 

8.9068735 

cu.  in.  in  1  (U.  S.)  gallon 

231. 

2.3636120 

ft.  Ib.  per  sec.  in  1  H.  P. 

550. 

2.7403627 

kg.  m.  per  sec.  in  1  H.  P. 

76.0404 

1.8810445 

watts  in  1  H.  P. 

745.957 

2.8727135 

Table  VI.     Degrees  to  Radians 


1° 

.01745 

10° 

.17453 

100° 

1.74533 

6' 

.00175 

6' 

.00003 

2° 

.03491 

20° 

.34907 

110° 

1.91986 

7' 

.00204 

7' 

.00003 

3° 

.05236 

30° 

.52360 

120° 

2.09440 

8' 

.00233 

8' 

.00004 

4° 

.06981 

40° 

.69813 

130° 

2.26893 

9' 

.00262 

9' 

.00004 

5° 

.08727 

50° 

.87266 

140° 

2.44346 

10' 

.00291 

10' 

.00005 

6° 

.10472 

60° 

1.04720 

150° 

2.61799 

20' 

.005S2 

20' 

.00010 

7° 

.12217 

70° 

1.22173 

160° 

2.79253 

30' 

.00873 

30' 

.00015 

8° 

.13963 

80° 

1.39626 

170° 

2.96706 

40' 

.01164 

40' 

.00019 

9° 

.15708 

90° 

1.57080 

180° 

3.14159 

50' 

.01454 

50' 

.00024 

327 


Table  VII.     Compound  Interest  Table 

Amount  of  One  Dollar  Principal  with  Compound  Interest  at  Various  Rates. 


a 
•< 
H 
P 

2}  Per 
Cent. 

3  Per 

Cent. 

3J  Per 
Cent. 

4  Per 
Cent. 

4.^  Per 
Cent. 

5  Per 

Cent. 

5J  Per 
Cent. 

6  Per 
Cent. 

6£  Per 
Cent. 

7  Per 
Cent. 

8  Per 
Cent. 

1 

$1.025 

$1.030 

$1.035 

$1.040 

$1.045 

$1.050 

$1.055 

$1.060 

$1.065 

$1.070 

$1.800 

2 

1.051 

1.061 

1.071 

1.082 

1.092 

1.103 

1.113 

1.124 

1.134 

1.145 

1.166 

3 

1.077 

1.093 

1.109 

1.125 

1.141 

1.158 

1.174 

1.191 

1.208 

1.225 

1.260 

4 

1.104 

1.126 

1.148 

1.170 

1.193 

1.216 

1.239 

1.262 

1.286 

1.311 

1.360 

5 

1.131 

1.159 

1.188 

'1.217 

1.246 

1.276 

1.307 

1.338 

1.370 

1.403 

1.469 

6 

1.160 

1.194 

1.229 

1.265 

1.302 

1.340 

1.379 

1.419 

1.459 

1.501 

1.587 

7 

1.189 

1.230 

1.272 

1.316 

1.361 

1.407 

1.455 

1.504 

1.554 

1.606 

1.714 

8 

1.218 

1.267    1.317 

1.369 

1.422 

1.477 

1.535 

1.594 

1.655 

1.718 

1.851 

9 

1.249 

1.305  1    1.363 

1.423 

1.486 

1.551 

1.619 

1.689 

1.763 

1.838 

1.999 

10 

1.280 

1.344 

1.411 

1.480 

1.553 

1.629 

1.708 

1.791 

1.877 

1.967 

2.159 

11 

1.312 

1.384 

1.460 

1.539 

1.623 

1.710 

1.802 

1.898 

1.999 

2.105 

2.332 

12 

1.345 

1.426 

1.511 

1.601 

1.696 

1.796 

1.901 

2.012 

2.129 

2.252 

2.518 

13 

1.379 

1.469 

1.564 

1.6G5 

1.772 

1.886 

2.006 

2.133 

2.267 

2.410 

2.720 

14 

1.413 

1.513 

1.619 

1.732 

1.852 

1.980 

2.116 

2.261 

2.415 

2.579 

2.937 

15 

1.448 

1.558 

1.675 

1.801 

1.935 

2.079 

2.232 

2.397 

2.572 

2.759 

3.172 

16 

1.485 

1.605 

1.734 

1.873 

2.022 

2.183 

2.355 

2.540 

2.739 

2.952 

3.426 

17 

1.522 

1.653 

1.795 

1.948 

2.113 

2.292 

2.485 

2.693 

2.917 

3.159 

3.700 

18 

1.560 

1.702 

1.857 

2.026 

2.208 

2.407 

2.621 

2.854 

3.107 

3.380 

3.996 

19 

1.599 

1.754 

1.923 

2.107 

2.308 

2.527 

2.766 

3.026 

3.309 

3.617 

4.316 

20 

1.639 

1.806 

1.990 

2.191 

2.412 

2.653 

2.918 

3.207 

3.524 

3.870 

4.661 

21 

1.680 

1.860 

2.059 

2.279 

2.520 

2.786 

3.078 

3.400 

3.753 

4.141 

5.034 

22 

1.722 

1.916 

2.132 

2.370 

2.634 

2.925 

3.248 

.3.604 

3.997 

4.430 

5.437 

23 

1.765 

1.974 

2.206 

2.465 

2.752 

3.072 

3.426 

3.820 

4.256 

4.741 

5.871 

24 

1.809 

2.033 

2.283 

2.563 

2.876 

3.225 

3.615 

4.049 

4.533 

5.072 

6.341 

25 

1.854 

2.094 

2.363 

2.666 

3.005 

3.386 

3.813 

4.292 

4.828 

5.427 

6.848 

26 

1.900 

2.157 

2.446 

2.772 

3.141 

3.556 

4.023 

4.549 

5.142 

5.807 

7.396 

27 

1.948 

2.221 

2.532 

2.883 

3.282 

3.733 

4.244 

4.822 

5.476 

6.214 

7.988 

28 

1.996 

2.288 

2.620 

2.999 

3.430 

3.920 

4.478 

5.112 

5.832 

6.649 

8.627 

29 

2.046 

2.357 

2.712 

3.119 

3.584 

4.116 

4.724 

5.418 

6.211 

7.114 

9.317 

30 

2.098 

2.427 

2.807 

3.243 

3.745 

4.322 

4.984 

5.743 

6.614 

7.612 

10.063 

31 

2.150 

2.500 

2.905 

3.373 

3.914 

4.538 

5.258 

6.088 

7.044 

8.145 

10.868 

32 

2.204 

2.575 

3.007 

3.508 

4.090 

4.765 

5.547 

6.453 

7.502 

8.715 

11.737 

33 

2.259 

2.652 

3.112 

3.648 

4.274 

5.003 

5.852 

6.841 

7.990 

9.325 

12.676 

34 

2.315 

2.732 

3.221 

3.794 

4.466 

5.253 

6.174 

7.251 

8.509 

9.978 

13.690 

35 

2.373 

2.814 

3.334 

3.946 

4.667 

5.516 

6.514 

7.686 

9.062 

10.677 

14.785 

36 

2.433 

2.898 

3.450 

4.104 

4.877 

5.792 

6.872 

8.147 

9.651 

11.424 

15.968 

37 

2.493 

2.985 

3.571 

4.268 

5.097 

6.081 

7.250 

8.636 

10.279 

12.224 

17.246 

38 

2.556 

3.075 

3.696 

4.439 

5.326 

6.385 

7.649 

9.154 

10.947 

13.079 

18.625 

39 

2.620 

3.167 

3.825 

4.616 

5.566 

6.705 

8.069 

9.704 

11.658 

13.995 

20.115 

40 

2.685 

3.262 

3.959 

4.801 

5.816 

7.040 

8.513 

10.286 

12.416 

14.974 

21.725 

41 

2.752 

3.360 

4.098 

4.993 

6.078 

7.392 

8.982 

10.903 

13.223 

16.023 

23.462 

42 

2.821 

3.461 

4.241 

5.193 

6.352 

7.762 

9.476 

11.557 

14.083 

17.144 

25.339 

43 

2.892 

3.565 

4.390 

5.400 

6.637 

8.150 

9.997 

12.250 

14.998 

18.344 

27.367 

44 

2.964 

3.671 

4.543 

5.617 

6.936 

8.557 

10.547 

12.985 

15.973 

19.628 

29.556 

45 

3.038 

3.782 

4.702 

5.841 

7.248 

8.985 

11.127 

13.765 

17.011 

21.002 

31.920 

46 

3.114 

3.895 

4.867 

6.075 

7.574 

9.434 

11.739 

14.590 

18.117 

22.473 

34.474 

47 

3.192 

4.012 

5.037 

6.318 

7.915 

9.906  12.384 

15.466 

19.294 

24.046 

37.232 

48 

3.271 

4.132 

5.214 

6.571 

8.271 

10.401   13.065 

16.394 

20.549 

25.729 

40.211 

49 

3.353 

4.256 

5.396 

6.833 

8.644 

10.921 

13.784 

17.378 

21.884 

27.530 

43.427 

50 

3.437 

4.384 

5.585 

7.107 

9.033 

11.467 

14.542 

18.420 

23.307 

29.457 

46.902 

328 


Table  VIII.     American  Experience  Mortality  Table 


Based  on  100,000  living  at  age  10. 


At 

Number 

At 

Number 

At 

Number 

At 

Number 

Age. 

Surviving. 

Deaths. 

Age. 

Surviving. 

Deaths. 

Age. 

Surviving. 

Deaths. 

Age. 

Surviving. 

Deaths. 

10 

100,000 

749 

35 

81,822 

732 

60 

57,917 

1,546 

85 

5,485 

1,292 

11 

99,251 

746 

36 

81,090 

737 

61 

56,371 

1,628 

86 

4,193 

1,114 

12 

98,505 

743 

37 

80,353 

742 

62 

54,743 

1,713 

87 

3,079 

933 

13 

97,762 

740 

38 

79,611 

749 

63 

53,030 

1,800 

88 

2,146 

744 

14 

97,022 

737 

39 

78,862 

756 

64 

51,230 

1,889 

89 

1,402 

555 

IS 

96,285 

735 

40 

78,106 

765 

65 

49,341 

1,980 

90 

847 

385 

16 

95,550 

732 

41 

77,341 

774 

66 

47,361 

2,070 

91 

462 

246 

17 

94,818 

729 

42 

76,567 

785 

67 

45,291 

2,158 

92 

216 

137 

18 

94,089 

727 

43 

75,782 

797 

68 

43,133 

2,243 

93 

79 

58 

19 

93,362 

725 

44 

74,985 

812 

69 

40,890 

2,321 

94 

21 

18 

20 

92,637 

723 

45 

74,173 

828 

70 

38,569 

2,391 

95 

3 

3 

21 

91,914 

722 

46 

73,345 

848 

71 

36,178 

2,448 

22 

91,192 

721 

47 

72,497 

870 

72 

33,730 

2,487 

23 

90,471 

720 

48 

71,627 

896 

73 

31,243 

2,505 

24 

89,751 

719 

49 

70,731 

927 

74 

28,738 

2,501 

25 

89,032 

718 

50 

69,804 

962 

75 

26,237 

2,476 

26 

88,314 

718 

51 

68,842 

1,001 

76 

23,761 

2,431 

27 

87,596 

718 

52 

67,841 

1,044 

77 

21,330 

2,369 

28 

86,878 

718 

53 

66,797 

1,091 

78 

18,961 

2,291 

29 

86,160 

719 

54 

65,706 

1,143 

79 

16,670 

2,196 

30 

85,441 

720 

55 

64,563 

1,199 

80 

14,474 

2,091 

31 

84,721 

721 

56 

63,364 

1,260 

81 

12,383 

1,964 

32 

84,000 

723 

57 

62,104 

1,325 

82 

10,419 

1,816 

33 

83,277 

726 

58 

60,779 

1,394 

83 

8,603 

1,648 

34 

82,551 

729 

59 

59,385 

1,468 

84 

6,955 

1,470 

329 


Table  IX.     Heights  and  Weights  of  Men 

Light-face  figures  are  20  per  cent,  under  and  over  the  average. 


AGES. 

S 

£ 

01 

I 

•* 

n 

i 

O 

n 
i 

IO 

w 

105 
131 

157 

1 

OJ 

I 

••# 

107 
134 

161 

s 
s 

107 
134 

161 

§ 
I 

10 

i< 
| 

cq 

OJ 
<N 

I 

C* 

•* 
« 

A 

123 
154 
185 

a 
3 

00 

5 

i 

OJ 

2 

•* 

129 
161 
193 

50-54 

o 

B 
1 

10 

130 
163 

196 

Ft. 
5 

In. 
0 

96 
120 

144 

100 
125 

150 

102 
128 
154 

106 
133 

160 

107 
134 
161 

Ft. 
5 

In. 
8 

117 
146 
175 

121 
151 
181 

126 
157 

188 

128 
160 

192 

130 
163 

196 

1 

98 
122 
146 

101 
126 
151 

103 
129 
155 

105 
131 

157 

107  109 
134  136 
161  163 

109  109 
136  136 
163  163 

9 

120  124  127 
150155  159 

180186191 

130  132 
162  165 
194  198 

133 
166 

199 

134 
167 

200 

134 
168 

202 



2 
3 
4 
5 

99 
124 
149 
102 
127 
152 

102 
128 
154 
105 
131 
157 

105 
131 
157 

106 
133 
160 
109 
136 
163 

109 
136 
163 

110 
138 
166 

113 
141 
169 

110 
138 
166 
113 
141 
169 

110 
138 
166 
113 
141 
169 

10 

11 

123 
154 
185 
127 
159 
191 

127  131 
159  164 

191  197 

134 
167 

200 

136  137 
170  171 
204205 

138 
172 
206 
142 
177 
212 

138 
173 
208 
142 
178 
214 

107 
134 
161 

111 
139 

167 

131 
164 

197 

135 
169 

203 

138 
173 
208 

140  142 
175  177 
210212 

105  108  110 
131  135  138 

157:162  166 

112 
140 

168 

114 
143 
172 

115  116 
144  145 
173  174 

116 
145 
174 

6 

0 

132  136  140  143 
165  170  175  179 
198  204  210  215 

144  146'  146  146 
180  183  182  183 

216220218220 

107 
134 
161 

110 
138 
166 

113 
141 
169 

114 
143 
172 
118 
147 
176 

117 
146 
175 

118 
147 
176 

119 
149 
179 

119 
149 
179 

1 

136142 
170  177 
204;212 

145  148 
181  185 
217  222 

149  151 
186  189 

223  227 

150  151 
188  189 

226  227 

6 

110114 
138  142 
166170 

iir, 
145 
174 

120 
150 
180 
124 
155 
186 

121 
151 
181 

122 
153 

184 

122 
153 
184 

2 

141  147 
176  184 
211  221 

150 
188 
226 

154  155!  157 
192  194  196 
230:233  235 

166 
194 

233 

155 
194 
233 

7 

114118 
142  147 

170  176 

i 

120 
150 

180 

122 
152 

182 

125 
156 

187 

126 
158 

190 

126 
158 
190 

3 

145 
181 

217 

152 
190 

228 

156 
195 
234 

160 
200 
240 

1621163 
203204 
244,245i 

161 

201 
241 

158 
198 
238 

330 


FOUR  PLACE  TABLES  33 J 

EXPLANATION  OF  TABLE  II* 
VALUES  AND  LOGARITHMS  OF  TRIGONOMETRIC  FUNCTIONS 

1.  DIRECT  READING  OF  THE  VALUES.  This  table  gives  the  sines, 
cosines,  tangents  and  cotangents  of  the  angles  from  0°  to  45°;  and  by 
a  simple  device,  indicated  by  the  printing,  the  values  of  these  functions 
for  angles  from  45°  to  90°  may  be  read  directly  from  the  same  table. 
For  angles  less  than  45°  read  down  the  page,  the  degrees  and  minutes 
being  found  on  the  left;  for  angles  greater  than  45°  read  up  the  page 
the  degrees  and  minutes  being  found  on  the  right. 

To  find  a  function  of  an  angle  (such  as  15°  27',  for  example)  we 
employ  the  process  of  interpolation.  To  illustrate,  let  us  find  tan  15° 
27'.  In  the  table  we  find  tan  15°  20'  =  .2742  and  tan  15°  30'  =  .2773; 
we  know  that  tan  15°  27'  lies  between  these  two  numbers.  The  process 
of  interpolation  depends  on  the  assumption  that  between  15°  20'  and 
15°  30'  the  tangent  of  the  angle  varies  directly  as  the  angle;  while  this 
assumption  is  not  strictly  true,  it  gives  an  approximation  sufficiently 
accurate  for  a  four-place  table.  Thus  we  should  assume  that  tan 
15°  25'  is  halfway  between  .2742  and  .2773.  We  may  state  the  problem 
as  follows:  An  increase  of  10'  in  the  angle  increases  the  tangent  .0031; 
assuming  that  the  tangent  varies  as  the  angle,  an  increase  of  7'  in  the 
angle  will  increase  the  tangent  by  .7  X  .0031  =  .00217.  Retaining  only 
four  places  we  write  this  .0022.  Hence 

tan  15°  27'  =  .2742  +  .0022  =  .2764. 

The  difference  between  two  successive  values  in  the  table  is  called 
the  tabular  difference  (.0031  above).  The  proportional  part  of  the 
tabular  difference  which  is  used  is  called  the  correction  (.0022  above), 
and  is  found  by  multiplying  the  tabular  difference  by  the  appropriate 
fraction  (.7  above). 

Example  1.     Find  sin  63°  52'. 

\Vefind 

sin  63°  50'  =  .8975. 
tabular  difference  =  .0013  (subtracted  mentally  from  the  table). 

correction  =  .2  X  .0013  =  .0003  (to  be  added). 
Hence, 

sin  63°  52'  =  .8978. 

*  The  use  of  Table  I.  is  explained  on  pages  80-86  of  the  text. 


332  MATHEMATICS 

Example  2.     Find  tan  37°  44'. 

tan  37°  40'  =  .7720 
tabular  difference  =  .0046 

correction  =  .4  X  .0046  =  .0018. 
Hence, 

tan  37°  44'  =  .7738. 

Example  3.     Find  cos  65°  24'. 

cos  65°  20'  =  .4173 
tabular  difference  =  26;  .4  X  26  =  10 

(to  be  subtracted  because  the  cosine  decreases  as  the  angle  increases). 
Hence 

cos  65°  24'  =  .4163. 
Example  4.     Find  ctn  32°  18'. 

ctn  32°  10'  =  1.5900 

tabular  difference  =  102;          .8  X  102  =  82  (to  be  subtracted). 
Hence, 

ctn  32°  18'  =  1.5818. 

Rule.  To  find  a  trigonometric  function  of  an  angle  by  interpolation: 
select  the  angle  in  the  table  which  is  next  smaller  than  the  given  angle,  and 
read  its  sine  (cosine,  tangent,  or  cotangent  as  the  case  may  be)  and  the 
tabular  difference.  Compute  the  correction  as  the  proper  proportional 
part  of  the  tabular  difference.  In  case  of  sines  or  tangents  ADD  the  cor- 
rection: in  case  of  cosines  or  cotangents,  SUBTRACT  it. 

2.  REVERSE  READINGS.  Interpolation  is  also  used  in  finding  the 
angle  when  one  of  its  functions  is  given. 

Example  1.     Given  sin  x  =  .3294,  to  find  x. 

Looking  in  the  table  we  find  the  sine  which  is  next  less  than  the  given 
sine  to  be  .3283,  and  this  belongs  to  19°  10'.  Subtract  the  value  of  the 
sine  selected  from  the  given  sine  to  obtain  the  actual  difference  =  .0011; 
note  that  the  tabular  difference  =  .0028.  We  may  state  the  problem 
as  follows:  an  increase  of  .0028  in  the  function  increases  the  angle  10'; 
then  aa  increase  of  .0011  in  the  function  will  increase  the  angle  11/28 
of  10  =  4  (to  be  added).  Hence  x  =  19°  14'. 

Example  2.     Given  cos  x  =  .2900,  to  find  x. 

The  cosine  in  the  table  next  less  than  this  is  .2896  and  belongs  to 
73°  10';  the  tabular  difference  is  28;  the  actual  difference  is  4;  correction 
=  4/28  of  10  =  1  (to  be  subtracted).  Hence  x  =  73°  9'. 


FOUR  PLACE  TABLES  333 

Rule.  To  find  an  angle  when  one  of  its  trigonometric  functions  is 
given:  select  from  the  table  the  same  named  function  which  is  next  less  than 
the  given  function,  noting  the  corresponding  angle  and  the  tabular  differ- 
ence: compute  the  actual  difference  (between  the  selected  value  of  the  func- 
tion and  the  given  value),  divide  it  by  the  tabular  difference,  and  multiply 
the  result  by  10;  this  gives  the  correction  which  is  to  be  added  if  the  given 
function  is  sine  or  tangent,  and  to  be  subtracted  if  the  given  function  is 
cosine  or  cotangent. 

3.    THE    LOGARITHMS    OF    THE    TRIGONOMETRIC    FUNCTIONS.      If    it    is 

required  to  find  log  sin  63°  52',  the  most  obvious  way  is  to  find  sin  63°  52' 
=  .8978,  and  then  to  find  in  Table  I,  log  .8978  =  9.9532  -  10,  but  this 
involves  consulting  two  tables.  To  avoid  the  necessity  of  doing  this, 
Table  II  gives  the  logarithms  of  the  sines,  cosines,  tangents,  and  co- 
tangents. The  student  should  note  that  the  sines  and  cosines  of  all 
acute  angles,  the  tangents  of  all  acute  angles  less  than  45°  and  the 
cotangents  of  all  acute  angles  greater  than  45°  are  proper  fractions, 
and  their  logarithms  end  with  —  10,  which  is  not  printed  in  the  table, 
but  which  should  be  written  down  whenever  such  a  logarithm  is  used. 

Example  1.     Find  log  sin  58°  24'. 

In  the  row  having  58°  20'  on  the  right  and  in  the  column  having 
sine  at  the  bottom  find  log  sin  58°  20'  =  9.9300  -  10;  the  tabular  differ- 
ence is  8;  correction  =  .4  X  8  =  3  (to  be  added).  Hence 

log  sin  58°  24'  =  9.9303  -  10. 

(In  case  of  sine  and  tangent  add  the  correction.) 
Example  2.     Find  log  cos  48°  38'. 

log  cos  48°  30'  =  9.8213  -  10,  tabular  difference  15; 
.8  X  15  =  12  (subtract)  therefore  log  cos  48°  38'  =  9.8201  -  10. 

(In  case  of  cosine  and  cotangent,  subtract  the  correction.) 

Example  3.     Given  log  tan  x  =  0.0263,  to  find  x. 

The  log  tan  in  Table  II  next  less  than  the  given  one  is  0.0253  and 
belongs  to  46°  40';  actual  difference  is  10;  tabular  difference  is  25; 
correction  =  10/25  of  10  =  4.  Hence  x  =  46°  44'. 

Example  4.     Given  log  cos  x  =  9.9726  —  10,  to  find  x. 

The  logarithmic  cosine  next  less  than  the  given  one  is  9.9725  —  10 
and  belongs  to  20°  10';  actual  difference  =  1;  tabular  difference  =  5; 
correction  =  1/5  X  10  =  2  (subtract).  Hence  x  =  20°  8'. 


INDEX 


Abscissa,  38 

Addition  formulas,  116 

Angles,  91,  94,  101,  104,  106 

trigonometric  functions  of, 

91,  102 
Annuity,  amount  of,  254,  2551 

cost  of,  259 

present  value  of,  254,  258 
Area,  by  offsets,  147 

by  rectangular  coordinates, 
147 

of  ellipse,  202 

of  triangle,  134 
Associative  law,  3 
Asymptote,  205,  206 
Auxiliary  circle,  202 
Average,  262 

arithmetic,  262,  263 

weighted  arithmetic,  263 

geometric,  265 
Axis,  38,  61,  196,  199 

Bearing,  140 

Binomial  coefficients,  8,  277 

series,  281 

theorem,  7,  276,  278 

Characteristic,  77 
Circle,  190 

auxiliary,  202 
Coefficients,  23 

binomial,  8,  277 

variability,  300 
Cologarithm,  84 
Combinations,  270,  273 
Commutative  law,  3 
Components,  of  a  force,  156 

rectangular,  157 
Compound    interest,    255,   256, 

286 

Conic  sections,  190 
Coordinates,  38 
Corners,  142 
Correlation,  304 

coefficient  of,  307 


Correlation,  measure  of,  305 

table,  306 
Cosecant,  92,  103 
Cosines,  91,  102 

law  of,  122 
Cotangent,  92,  103 
Couple,  168 
Crane,  160 

Degree,  23 

Deviation,  standard,  298 
Diagrams,  41,  47,  49,  50,  51 
Directrix,  195 

Distance  between  two  points,  53 
Distributive  law,  3 
Division,  point  of,  55 
ratio  of,  54 

Eccentricity,  of  ellipse,  200 

of  hyperbola,  204 
Elimination,  17 
Ellipse,  190,  199 

area  of,  202 
Equation,  of  a  circle,  190 

of  a  curve,  58 

of  an  ellipse,  201 

of  a  hyperbola,  204,  205, 
206 

of  a  parabola,  195,  197 

of  a  straight  line,  63,  64,  66, 

67 
Equations,  definition  of,  11 

conditional,  13 

empirical,  226 

equivalent,  14 

general,  68,  191 

having  given  roots,  32 

in  quadratic  form,  29 

linear,  18,  24,  68 

quadratic,  24 

simultaneous,  16 

trigonometric,  107 

transformation  of,  14 
Equilibrium,  conditions  of,  169 
Error,  177,  181 


335 


336 


INDEX 


Error,  curve  of,  298 

in  a  fraction,  179 

in  parts  of  a  triangle,  185 

in  a  product,  178 

in  a  square,  183 

in  a  square  root,  183 

in  a  sum,  178 

in  trigonometric  functions, 
183 

probable,  300 
Evolution,  4 
Expectation,  292 
Exponents,  4,  5,  6,  23 

Focal  properties,  211,  213 
Focus,  195,  199,  201,  204 
Forces,  components  of,  156 
concentrated,  154 
distributed,  154 
graphical  representation  of, 

154 

moments  of,  167 
parallelogram  of,  155 
resolution  of,  156 
Frequency  distribution  curves, 

296 
Functions,  218 

of  complementary  angles,  93 
of  half  an  angle,  118 
of  negative  angles,  110 
of  twice  an  angle,  118 
periodic,  110 
trigonometric,  91,  102,  105, 

109 
Fundamental  relations,  95,  104 

Graphical  solution,  36,  99,  154, 

158,  220,  228,  238 
Graphs,  41,  113,  154,  220 

Hyperbola,  190,  204 

equilateral   or  rectangular, 
205 

Identities,  12 
Imaginary  numbers,  30 
Intercepts,  61 
Interpolation,  81,  331 
Intersection,  points  of,  62 

of  conies,  215 

of  loci,  62,  208 


Involution,  3 
Irrational  numbers,  2 

Latus  rectum,  196 
Law,  associative,  3 

commutative,  3 

distributive,  3 

of  cosines,  122 

of  exponents,  6 

of  sines,  121 

of  tangents,  124 
Lines,  base,  140 

bearing  of,  140 

parallel,  63,  67 

perpendicular,  67 

random,  146 

range,  140 

slope  of,  66 

through  the  origin,  64 

through  two  points,  66 

township,  140 
Locus,  of  a  point,  57 

of  an  equation,  58,  59 
Logarithmic  paper,  239 

plotting,  237 
Logarithms,  Briggs,  76 

computation  by,  87 

computation  of,  76 

definition  of,  72 

Napierian,  76 

properties  of,  74 

Mantissa,  77 

Mass,  153 

Mean,  arithmetic,  245,  262,  263 

geometric,  249,  265 
Measurement,  1,  181 

on  level  ground,  143 

on  slopes,  144 

of  force,  154 
Median,  264 
Mendel's  law,  282 
Middle  point,  56 
Mode,  264 
Moments,  of  force,  167 

center  of,  167 

composition  of,  167 
Momentum,  153 

Normal,  211 
Number,  2,  3,  30,  31 


INDEX 


337 


Oblique   triangles,   solution   of, 

120,  125 
Offsets,  144 
Ordinate,  38 
Origin,  38 

Parabola,  190,  195 
Parallelogram  of  forces,  155 
Periodic  functions,  110 
Permutations,  270 
Perpetuity,  260 
Point,  of  division,  55 

of  intersection,  62 
Polygon  of  forces,  162 
Polynomial,  22 
Principal  meridian,  140 
Probability,  291 

curve,  297,  298 

Probable     error,     in     a     single 
measurement,  300 

of  arithmetic  average,  300 

of  standard  deviation,  301 
Progression,  arithmetic,  243 

geometric,  247,  252 
Proportional  quantities,  64,  219, 
220 

Quadrantal  angles,  104 
Quadratic  equation,  24 

kind  of  roots,  30,  33 

number  of  roots,  33 

solution  of,  26,  27 

sum  and  product  of  roots, 
32 

Radian,  113 
Ratio  of  division,  54 
Rational  number,  2 
Rectangular  components,  157 
coordinates,  38 


Rectangular  hyperbola,  205,  206 

Regression  curve,  309 

Resolution  of  forces,  156 

Resultant,  155 

of  concurrent  forces,  163 
of  parallel  forces,  165 

Right  triangles,  solution  of,  97 

Root,  12 

Scales,  36 
Secant,  92,  102 
Series,  binomial,  281 

infinite  geometric,  252 
Sines,  91,  102 

law  of,  121 
Slide  rule,  88 
Slope,  66 

Statistical  data,  40 
Substitution,  12,  233 
Symmetry,  61 

Tabular  difference,  81,  331 
Tangents,  91,  102,  211 

law  of,  124 

Translation  of  axes,  194 
Triangle,  oblique,  120,  125 

of  forces,  158 

right,  97 

Trigonometric  functions,  of  an 
acute  angle,  91 

of  any  angle,  102 

graphs  of,  109 

line  representation  of,  105 

Variable,  218 
Variation,  219 

direct,  219 

inverse,  219 

joint,  219 

constant  of,  219,  222 


Printed  in  the  United  States  of  America. 


TRIGONOMETRY 

BY 

ALFRED  MONROE  KENYON 

PROFESSOR  OF  MATHEMATICS,  PURDUE  UNIVERSITY 

AND  LOUIS  INGOLD 

ASSISTANT  PROFESSOR  OF  MATHEMATICS,  THE  UNIVERSITY  OF 
MISSOURI 

Edited  by  EARLE  RAYMOND  HEDRICK 

Trigonometry,  flexible  cloth,  pocket  size,  long  I2mo  (xi-srij2  pp.)  with  Complete 

Tables  (xviii  + 124 pp.),  $t-5O 

Trigonometry  (xi-\- 132  pp.)  with  Brief  Tables  (xviii -{- 12  pp.),  $1.20 
Macmillan  Logarithmic  and  Trigonometric  Tables,  flexible  cloth,  pocket  size,  long 

izmo  (xviii  +  124  pp.) ,  $60 

FROM   THE   PREFACE 

The  book  contains  a  minimum  of  purely  theoretical  matter.  Its  entire 
organization  is  intended  to  give  a  clear  view  of  the  meaning  and  the  imme- 
diate usefulness  of  Trigonometry.  The  proofs,  however,  are  in  a  form  that 
will  not  require  essential  revision  in  the  courses  that  follow.  .  .  . 

The  number  of  exercises  is  very  large,  and  the  traditional  monotony  is 
broken  by  illustrations  from  a  variety  of  topics.  Here,  as  well  as  in  the  text, 
the  attempt  is  often  made  to  lead  the  student  to  think  for  himself  by  giving 
suggestions  rather  than  completed  solutions  or  demonstrations. 

The  text  proper  is  short;  what  is  there  gained  in  space  is  used  to  make  the 
tables  very  complete  and  usable.  Attention  is  called  particularly  to  the  com- 
plete and  handily  arranged  table  of  squares,  square  roots,  cubes,  etc. ;  by  its 
use  the  Pythagorean  theorem  and  the  Cosine  Law  become  practicable  for 
actual  computation.  The  use  of  the  slide  rule  and  of  four-place  tables  is 
encouraged  for  problems  that  do  not  demand  extreme  accuracy. 

Only  a  few  fundamental  definitions  and  relations  in  Trigonometry  need  be 
memorized;  these  are  here  emphasized.  The  great  body  of  principles  and 
processes  depends  upon  these  fundamentals;  these  are  presented  in  this  book, 
as  they  should  be  retained,  rather  by  emphasizing  and  dwelling  upon  that 
dependence.  Otherwise,  the  subject  can  have  no  real  educational  value,  nor 
indeed  any  permanent  practical  value. 


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ELEMENTARY  MATHEMATICAL 
ANALYSIS 

BY 

JOHN  WESLEY  YOUNG 

PROFESSOR  OF  MATHEMATICS  IN  DARTMOUTH  COLLEGE 

AND  FRANK  MILLETT  MORGAN 

ASSISTANT  PROFESSOR  OF  MATHEMATICS  IN  DARTMOUTH  COLLEGE 


Edited  by   EARLE    RAYMOND    HEDRICK,   Professor  of  Mathematics 
in    the   University   of  Missouri 

Cloth,  i2tno,  542  pp.,  $2.60 

A  textbook  for  the  freshman  year  in  colleges,  universities,  and 
technical  schools,  giving  a  unified  treatment  of  the  essentials  of 
trigonometry,  college  algebra,  and  analytic  geometry,  and  intro- 
ducing the  student  to  the  fundamental  conceptions  of  calculus. 

The  various  subjects  are  unified  by  the  great  centralizing 
theme  of  functionality  so  that  each  subject,  without  losing  its 
fundamental  character,  is  shown  clearly  in  its  relationship  to  the 
others,  and  to  mathematics  as  a  whole. 

More  emphasis  is  placed  on  insight  and  understanding  of 
fundamental  conceptions  and  modes  of  thought ;  less  emphasis 
on  algebraic  technique  and  facility  of  manipulation.  Due  recog- 
nition is  given  to  the  cultural  motive  for  the  study  of  mathe- 
matics and  to  the  disciplinary  value. 

The  text  presupposes  only  the  usual  entrance  requirements  in 
elementary  algebra  and  plane  geometry. 


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ANALYTIC  GEOMETRY 

BY 

ALEXANDER   ZIWET 

PROFESSOR  OF  MATHEMATICS  IN  THE  UNIVERSITY  OF  MICHIGAN 

AND  LOUIS  ALLEN  HOPKINS 

INSTRUCTOR  IN  MATHEMATICS  IN  THE  UNIVERSITY  OF  MICHIGAN 

Edited  by  EARLE  RAYMOND  HEDRICK 

Flexible  cloth.  111.,  izmo,  viii  +  j6g pp.,  $fj6o 

Combines  with  analytic  geometry  a  number  of  topics,  tradi- 
tionally treated  in  college  algebra,  that  depend  upon  or  are 
closely  associated  with  geometric  representation.  If  the  stu- 
dent's preparation  in  elementary  algebra  has  been  good,  this 
book  contains  sufficient  algebraic  material  to  enable  him  to 
omit  the  usual  course  in  College  Algebra  without  essential 
harm.  On  the  other  hand,  the  book  is  so  arranged  that,  for 
those  students  who  have  a  college  course  in  algebra,  the  alge- 
braic sections  may  either  be  omitted  entirely  or  used  only  for 
review.  The  book  contains  a  great  number  of  fundamental 
applications  and  problems.  Statistics  and  elementary  laws  of 
Physics  are  introduced  early,  even  before  the  usual  formulas 
for  straight  lines.  Polynomials  and  other  simple  explicit  func- 
tions are  dealt  with  before  the  more  complicated  implicit  equa- 
tions, with  the  exception  of  the  circle,  which  is  treated  early. 
The  representation  of  functions  is  made  more  prominent  than 
the  study  of  the  geometric  properties  of  special  curves.  Purely 
geometric  topics  are  not  neglected. 


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Publishers  64-66  Fifth  Avenue  New  Tork 


Analytic  Geometry  and  Principles  of  Algebra 

BY 

ALEXANDER   ZIWET 

PROFESSOR  OF  MATHEMATICS,  THE  UNIVERSITY  OF  MICHIGAN 

AND  LOUIS  ALLEN   HOPKINS 

INSTRUCTOR  IN  MATHEMATICS,  THE  UNIVERSITY  OF  MICHIGAN 

Edited  by  EARLE  RAYMOND   HEDRICK 

Cloth,  viii  +  369  pp.,  appendix,  answers,  index,  I2mo,  $i-7j 

This  work  combines  with  analytic  geometry  a  number  of  topics  traditionally 
treated  in  college  algebra  that  depend  upon  or  are  closely  associated  with 
geometric  sensation.  Through  this  combination  it  becomes  possible  to  show 
the  student  more  directly  the  meaning  and  the  usefulness  of  these  subjects. 

The  idea  of  coordinates  is  so  simple  that  it  might  (and  perhaps  should)  be 
explained  at  the  very  beginning  of  the  study  of  algebra  and  geometry.  Real 
analytic  geometry,  however,  begins  only  when  the  equation  in  two  variables 
is  interpreted  as  defining  a  locus.  This  idea  must  be  introduced  very  gradu- 
ally, as  it  is  difficult  for  the  beginner  to  grasp.  The  familiar  loci,  straight 
line  and  circle,  are  therefore  treated  at  great  length. 

In  the  chapters  on  the  conic  sections  only  the  most  essential  properties  of 
these  curves  are  given  in  the  text ;  thus,  poles  and  polars  are  discussed  only 
in  connection  with  the  circle. 

The  treatment  of  solid  analytic  geometry  follows  the  more  usual  lines.  But, 
in  view  of  the  application  to  mechanics,  the  idea  of  the  vector  is  given  some 
prominence;  and  the  representation  of  a  function  of  two  variables  by  contour 
lines  as  well  as  by  a  surface  in  space  is  explained  and  illustrated  by  practical 
examples. 

The  exercises  have  been  selected  with  great  care  in  order  not  only  to  fur- 
nish sufficient  material  for  practice  in  algebraic  work  but  also  to  stimulate 
independent  thinking  and  to  point  out  the  applications  of  the  theory  to  con- 
crete problems.  The  number  of  exercises  is  sufficient  to  allow  the  instructor 
to  make  a  choice. 

To  reduce  the  course  presented  in  this  book  to  about  half  its  extent,  the 
parts  of  the  text  in  small  type,  the  chapters  on  solid  analytic  geometry,  and 
the  more  difficult  problems  throughout  may  be  omitted. 


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Publishers  64-66  Fifth  Avenue  New  York 


THE  CALCULUS 

BY 

ELLERY  WILLIAMS  DAVIS 

PROFESSOR  OF  MATHEMATICS,  THE  UNIVERSITY  OF  NEBRASKA 

Assisted  by  WILLIAM  CHARLES  BRENKE,  Associate   Professoi    ol 
Mathematics,  the  University  of  Nebraska 

Edited  by  EARLE  RAYMOND  HEDRICK 

Cloth,  semi-flexible,  xxi  +  3&3  PP-  +  Tables  (63),  szmo,  $2.10 
Edition  De  Luxe,  flexible  leather  binding,  India  paper,  $2.30 

This  book  presents  as  many  and  as  varied  applications  of  the  Calculus 
as  it  is  possible  to  do  without  venturing  into  technical  fields  whose  subject 
matter  is  itself  unknown  and  incomprehensible  to  the  student,  and  without 
abandoning  an  orderly  presentation  of  fundamental  principles. 

The  same  general  tendency  has  led  to  the  treatment  of  topics  with  a  view 
toward  bringing  out  their  essential  usefulness.  Rigorous  forms  of  demonstra- 
tion are  not  insisted  upon,  especially  where  the  precisely  rigorous  proofs 
would  be  beyond  the  present  grasp  of  the  student.  Rather  the  stress  is  laid 
upon  the  student's  certain  comprehension  of  that  which  is  done,  and  his  con- 
yiction  that  the  results  obtained  are  both  reasonable  and  useful.  At  the 
same  time,  an  effort  has  been  made  to  avoid  those  grosser  errors  and  actual 
misstatements  of  fact  which  have  often  offended  the  teacher  in  texts  otherwise 
attractive  and  teachable. 

Purely  destructive  criticism  and  abandonment  of  coherent  arrangement 
are  just  as  dangerous  as  ultra-conservatism.  This  book  attempts  to  preserve 
the  essential  features  of  the  Calculus,  to  give  the  student  a  thorough  training 
in  mathematical  reasoning,  to  create  in  him  a  sure  mathematical  imagination, 
and  to  meet  fairly  the  reasonable  demand  for  enlivening  and  enriching  the 
subject  through  applications  at  the  expense  of  purely  formal  work  that  con- 
tains no  essential  principle. 


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Publisher!  64-66  Fifth  Avenue  Hew  Tork 


GEOMETRY 

BY 
WALTER  BURTON   FORD 

JUNIOR  PROFESSOR  OF  MATHEMATICS  IN  THE  UNIVERSITY  OF 
MICHIGAN 

AND  CHARLES   AMMERMAN 

THE  WILLIAM  McKiNLEY  HIGH  SCHOOL,  ST.  Louis 

Edited  by  EARLE  RAYMOND  HEDRICK,  Professor  of  Mathematics 

in  the  University  of  Missouri 

Plane  and  Solid  Geometry,  doth,  izmo,  319  pp.,  $125 
Plane  Geometry,  cloth,  I2mo,  213  pp.,  $  .80 
Solid  Geometry,  doth,  I2mo,  106  pp.,  $  .80 

STRONG   POINTS 

I.  The  authors  and  the  editor  are  well  qualified  by  training  and  experi- 
ence to  prepare  a  textbook  on  Geometry. 

II.  As   treated   in   this   book,  geometry   functions   in   the   thought   of  the 
pupil.     It  means  something  because  its  practical  applications  are  shown. 

III.  The  logical  as  well  as  the  practical  side  of  the  subject  is  emphasized. 

IV.  The  arrangement  of  material  is  pedagogical. 

V.  Basal  theorems  are  printed  in  black-face  type. 

VI.  The  book  conforms  to  the  recommendations  of  the  National  Com- 
mittee on  the  Teaching  of  Geometry. 

VII.  Typography  and  binding  are  excellent.     The  latter  is  the  reenforced 
tape  binding  that  is  characteristic  of  Macmillan  textbooks. 

"Geometry  is  likely  to  remain  primarily  a  cultural,  rather  than  an  informa- 
tion subject,""  say  the  authors  in  the  preface.  "  But  the  intimate  connection 
of  geometry  with  human  activities  is  evident  upon  every  hand,  and  constitutes 
fully  as  much  an  integral  part  of  the  subject  as  does  its  older  logical  and 
scholastic  aspect."  This  connection  with  human  activities,  this  application 
of  geometry  to  real  human  needs,  is  emphasized  in  a  great  variety  of  problems 
and  constructions,  so  that  theory  and  application  are  inseparably  connected 
throughout  the  book. 

These  illustrations  and  the  many  others  contained  in  the  book  will  be  seen 
to  cover  a  wider  range  than  is  usual,  even  in  books  that  emphasize  practical 
applications  to  a  questionable  extent.  This  results  in  a  better  appreciation 
of  the  significance  of  the  subject  on  the  part  of  the  student,  in  that  he  gains  a 
truer  conception  of  the  wide  scope  of  its  application. 

The  logical  as  well  as  the  practical  side  of  the  subject  is  emphasized. 

Definitions,  arrangement,  and  method  of  treatment  are  logical.  The  defi- 
nitions are  particularly  simple,  clear,  and  accurate.  The  traditional  manner 
of  presentation  in  a  logical  system  is  preserved,  with  due  regard  for  practical 
applications.  Proofs,  both  foimal  and  informal,  are  strictly  logical. 


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SLIDE-RULE 


r 


(1)  (*)  (3) 


I  la' 


LLL.I 


DIRECTIONS 

A  reasonably  accurate  slide-rule 
may  be  made  by  the  student,  for 
temporary  practice,  as  follows. 
Take  three  strips  of  heavy  stiff 
cardboard  1".3  wide  by  6"  long; 
these  are  shown  in  cross-section  in 
(1),  (2),  (3)  above.  On  (3) 
paste  or  glue  the  adjoining  cut 
of  the  slide  rule.  Then  cut  strips 
(2)  and  (3)  accurately  along  the 
lines  marked.  Paste  or  glue  the 
pieces  together  as  shown  in  (4) 
and  (5).  Then  (5)  forms  the 
slide  of  the  slide-rule,  and  it  will 
fit  in  the  groove  in  (4)  if  the  work 
has  been  carefully  done.  Trim 
off  the  ends  as  shown  in  the  large 
cut. 


m     o 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 

Los  Angeles 
This  book  is  DUE  on  the  last  date  stamped  below. 


RHTD  LD-UR0 

FEB161971 

16 1971 


Form  L9-Series  444 


UC  SOUTHERN  REGIONAL  LIBRARY  FACILITY 


A    000933189    3 


